[{"data":1,"prerenderedAt":3250},["ShallowReactive",2],{"subject-categories-math":3,"formula-de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fumkreisradius":546,"formula-related-de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fumkreisradius":3237},[4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,173,174,175,176,177,178,179,180,181,182,183,184,185,186,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545],{"category":5},"algebra",{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":65},"arithmetic",{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":101},"calculus",{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":172},"differential-equations",{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":187},"discrete-math",{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":208},"functions",{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":247},"geometry",{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":312},"linear-algebra",{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":368},"multivariable-calculus",{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":391},"numerics",{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":412},"sequences-series",{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":441},"stochastics",{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":518},"trigonometry",{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"id":547,"title":548,"aliases":549,"body":554,"category":247,"checks":3106,"conditions":3124,"conditionsHtml":3125,"deprecated":3126,"description":3099,"extension":3127,"figure":3128,"figureCaption":3129,"figureHtml":3199,"formulaId":3200,"latex":3201,"latexHtml":3202,"level":3203,"meta":3204,"navigation":3150,"path":3205,"redirectFrom":3206,"related":3207,"seo":3212,"stem":3213,"subject":596,"summary":3214,"tags":3215,"variables":3221,"__hash__":3236},"formulas_de\u002Fde\u002Fformulas\u002Fmath\u002Fgeometry\u002Fumkreisradius.md","Umkreisradius eines Dreiecks",[550,551,552,553],"Umkreis Dreieck","Radius Umkreis berechnen","R = abc\u002F4A","Umkreisradius gleichseitiges Dreieck",{"type":555,"value":556,"toc":3098},"minimark",[557,562,575,582,975,978,1010,1014,1237,1569,1573,1769,2075,2078,2211,2659,3011,3015],[558,559,561],"h2",{"id":560},"erklärung","Erklärung",[563,564,565,566,570,571,574],"p",{},"Der ",[567,568,569],"strong",{},"Umkreis"," eines Dreiecks geht durch alle drei Ecken. Sein Mittelpunkt ist der Schnittpunkt der drei ",[567,572,573],{},"Mittelsenkrechten",". Bei spitzwinkligen Dreiecken liegt er innen, bei rechtwinkligen auf der Mitte der Hypotenuse, bei stumpfwinkligen außerhalb.",[563,576,577,578,581],{},"Die erste Form ist der ",[567,579,580],{},"erweiterte Sinussatz",": Für jedes Paar aus Seite und gegenüberliegendem Winkel gilt",[583,584,587],"span",{"className":585},[586],"katex-display",[583,588,591,672],{"className":589},[590],"katex",[583,592,595],{"className":593},[594],"katex-mathml",[596,597,600],"math",{"xmlns":598,"display":599},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML","block",[601,602,603,667],"semantics",{},[604,605,606,625,628,642,644,658,660,664],"mrow",{},[607,608,609,613],"mfrac",{},[610,611,612],"mi",{},"a",[604,614,615,618,622],{},[610,616,617],{},"sin",[619,620,621],"mo",{},"⁡",[610,623,624],{},"α",[619,626,627],{},"=",[607,629,630,633],{},[610,631,632],{},"b",[604,634,635,637,639],{},[610,636,617],{},[619,638,621],{},[610,640,641],{},"β",[619,643,627],{},[607,645,646,649],{},[610,647,648],{},"c",[604,650,651,653,655],{},[610,652,617],{},[619,654,621],{},[610,656,657],{},"γ",[619,659,627],{},[661,662,663],"mn",{},"2",[610,665,666],{},"R",[668,669,671],"annotation",{"encoding":670},"application\u002Fx-tex","\\frac{a}{\\sin \\alpha} = \\frac{b}{\\sin \\beta} = \\frac{c}{\\sin \\gamma} = 2R",[583,673,677,789,876,961],{"className":674,"ariaHidden":676},[675],"katex-html","true",[583,678,681,686,778,782,786],{"className":679},[680],"base",[583,682],{"className":683,"style":685},[684],"strut","height:1.7936em;vertical-align:-0.686em;",[583,687,690,695,774],{"className":688},[689],"mord",[583,691],{"className":692},[693,694],"mopen","nulldelimiter",[583,696,698],{"className":697},[607],[583,699,703,765],{"className":700},[701,702],"vlist-t","vlist-t2",[583,704,707,760],{"className":705},[706],"vlist-r",[583,708,712,737,748],{"className":709,"style":711},[710],"vlist","height:1.1076em;",[583,713,715,720],{"style":714},"top:-2.314em;",[583,716],{"className":717,"style":719},[718],"pstrut","height:3em;",[583,721,723,727,732],{"className":722},[689],[583,724,617],{"className":725},[726],"mop",[583,728],{"className":729,"style":731},[730],"mspace","margin-right:0.1667em;",[583,733,624],{"className":734,"style":736},[689,735],"mathnormal","margin-right:0.0037em;",[583,738,740,743],{"style":739},"top:-3.23em;",[583,741],{"className":742,"style":719},[718],[583,744],{"className":745,"style":747},[746],"frac-line","border-bottom-width:0.04em;",[583,749,751,754],{"style":750},"top:-3.677em;",[583,752],{"className":753,"style":719},[718],[583,755,757],{"className":756},[689],[583,758,612],{"className":759},[689,735],[583,761,764],{"className":762},[763],"vlist-s","​",[583,766,768],{"className":767},[706],[583,769,772],{"className":770,"style":771},[710],"height:0.686em;",[583,773],{},[583,775],{"className":776},[777,694],"mclose",[583,779],{"className":780,"style":781},[730],"margin-right:0.2778em;",[583,783,627],{"className":784},[785],"mrel",[583,787],{"className":788,"style":781},[730],[583,790,792,796,867,870,873],{"className":791},[680],[583,793],{"className":794,"style":795},[684],"height:2.2519em;vertical-align:-0.8804em;",[583,797,799,802,864],{"className":798},[689],[583,800],{"className":801},[693,694],[583,803,805],{"className":804},[607],[583,806,808,855],{"className":807},[701,702],[583,809,811,852],{"className":810},[706],[583,812,815,833,841],{"className":813,"style":814},[710],"height:1.3714em;",[583,816,817,820],{"style":714},[583,818],{"className":819,"style":719},[718],[583,821,823,826,829],{"className":822},[689],[583,824,617],{"className":825},[726],[583,827],{"className":828,"style":731},[730],[583,830,641],{"className":831,"style":832},[689,735],"margin-right:0.0528em;",[583,834,835,838],{"style":739},[583,836],{"className":837,"style":719},[718],[583,839],{"className":840,"style":747},[746],[583,842,843,846],{"style":750},[583,844],{"className":845,"style":719},[718],[583,847,849],{"className":848},[689],[583,850,632],{"className":851},[689,735],[583,853,764],{"className":854},[763],[583,856,858],{"className":857},[706],[583,859,862],{"className":860,"style":861},[710],"height:0.8804em;",[583,863],{},[583,865],{"className":866},[777,694],[583,868],{"className":869,"style":781},[730],[583,871,627],{"className":872},[785],[583,874],{"className":875,"style":781},[730],[583,877,879,883,952,955,958],{"className":878},[680],[583,880],{"className":881,"style":882},[684],"height:1.988em;vertical-align:-0.8804em;",[583,884,886,889,949],{"className":885},[689],[583,887],{"className":888},[693,694],[583,890,892],{"className":891},[607],[583,893,895,941],{"className":894},[701,702],[583,896,898,938],{"className":897},[706],[583,899,901,919,927],{"className":900,"style":711},[710],[583,902,903,906],{"style":714},[583,904],{"className":905,"style":719},[718],[583,907,909,912,915],{"className":908},[689],[583,910,617],{"className":911},[726],[583,913],{"className":914,"style":731},[730],[583,916,657],{"className":917,"style":918},[689,735],"margin-right:0.0556em;",[583,920,921,924],{"style":739},[583,922],{"className":923,"style":719},[718],[583,925],{"className":926,"style":747},[746],[583,928,929,932],{"style":750},[583,930],{"className":931,"style":719},[718],[583,933,935],{"className":934},[689],[583,936,648],{"className":937},[689,735],[583,939,764],{"className":940},[763],[583,942,944],{"className":943},[706],[583,945,947],{"className":946,"style":861},[710],[583,948],{},[583,950],{"className":951},[777,694],[583,953],{"className":954,"style":781},[730],[583,956,627],{"className":957},[785],[583,959],{"className":960,"style":781},[730],[583,962,964,968,971],{"className":963},[680],[583,965],{"className":966,"style":967},[684],"height:0.6833em;",[583,969,663],{"className":970},[689],[583,972,666],{"className":973,"style":974},[689,735],"margin-right:0.0077em;",[563,976,977],{},"Die zweite Form braucht nur die Seiten und die Fläche, die man z.B. mit dem Satz des Heron berechnet.",[563,979,980,981,1009],{},"Beim rechtwinkligen Dreieck ist ",[583,982,984,997],{"className":983},[590],[583,985,987],{"className":986},[594],[596,988,989],{"xmlns":598},[601,990,991,995],{},[604,992,993],{},[610,994,666],{},[668,996,666],{"encoding":670},[583,998,1000],{"className":999,"ariaHidden":676},[675],[583,1001,1003,1006],{"className":1002},[680],[583,1004],{"className":1005,"style":967},[684],[583,1007,666],{"className":1008,"style":974},[689,735]," die halbe Hypotenuse. Das ist der Satz des Thales.",[558,1011,1013],{"id":1012},"herleitung","Herleitung",[563,1015,1016,1017,1176,1177,1206,1207,1236],{},"Setzt man die Flächenformel ",[583,1018,1020,1054],{"className":1019},[590],[583,1021,1023],{"className":1022},[594],[596,1024,1025],{"xmlns":598},[601,1026,1027,1051],{},[604,1028,1029,1032,1034,1041,1043,1045,1047,1049],{},[610,1030,1031],{},"A",[619,1033,627],{},[607,1035,1036,1039],{},[661,1037,1038],{},"1",[661,1040,663],{},[610,1042,632],{},[610,1044,648],{},[610,1046,617],{},[619,1048,621],{},[610,1050,624],{},[668,1052,1053],{"encoding":670},"A = \\frac{1}{2} bc \\sin \\alpha",[583,1055,1057,1075],{"className":1056,"ariaHidden":676},[675],[583,1058,1060,1063,1066,1069,1072],{"className":1059},[680],[583,1061],{"className":1062,"style":967},[684],[583,1064,1031],{"className":1065},[689,735],[583,1067],{"className":1068,"style":781},[730],[583,1070,627],{"className":1071},[785],[583,1073],{"className":1074,"style":781},[730],[583,1076,1078,1082,1158,1161,1164,1167,1170,1173],{"className":1077},[680],[583,1079],{"className":1080,"style":1081},[684],"height:1.1901em;vertical-align:-0.345em;",[583,1083,1085,1088,1155],{"className":1084},[689],[583,1086],{"className":1087},[693,694],[583,1089,1091],{"className":1090},[607],[583,1092,1094,1146],{"className":1093},[701,702],[583,1095,1097,1143],{"className":1096},[706],[583,1098,1101,1120,1128],{"className":1099,"style":1100},[710],"height:0.8451em;",[583,1102,1104,1107],{"style":1103},"top:-2.655em;",[583,1105],{"className":1106,"style":719},[718],[583,1108,1114],{"className":1109},[1110,1111,1112,1113],"sizing","reset-size6","size3","mtight",[583,1115,1117],{"className":1116},[689,1113],[583,1118,663],{"className":1119},[689,1113],[583,1121,1122,1125],{"style":739},[583,1123],{"className":1124,"style":719},[718],[583,1126],{"className":1127,"style":747},[746],[583,1129,1131,1134],{"style":1130},"top:-3.394em;",[583,1132],{"className":1133,"style":719},[718],[583,1135,1137],{"className":1136},[1110,1111,1112,1113],[583,1138,1140],{"className":1139},[689,1113],[583,1141,1038],{"className":1142},[689,1113],[583,1144,764],{"className":1145},[763],[583,1147,1149],{"className":1148},[706],[583,1150,1153],{"className":1151,"style":1152},[710],"height:0.345em;",[583,1154],{},[583,1156],{"className":1157},[777,694],[583,1159,632],{"className":1160},[689,735],[583,1162,648],{"className":1163},[689,735],[583,1165],{"className":1166,"style":731},[730],[583,1168,617],{"className":1169},[726],[583,1171],{"className":1172,"style":731},[730],[583,1174,624],{"className":1175,"style":736},[689,735]," in die zweite Form ein, kürzen sich ",[583,1178,1180,1193],{"className":1179},[590],[583,1181,1183],{"className":1182},[594],[596,1184,1185],{"xmlns":598},[601,1186,1187,1191],{},[604,1188,1189],{},[610,1190,632],{},[668,1192,632],{"encoding":670},[583,1194,1196],{"className":1195,"ariaHidden":676},[675],[583,1197,1199,1203],{"className":1198},[680],[583,1200],{"className":1201,"style":1202},[684],"height:0.6944em;",[583,1204,632],{"className":1205},[689,735]," und ",[583,1208,1210,1223],{"className":1209},[590],[583,1211,1213],{"className":1212},[594],[596,1214,1215],{"xmlns":598},[601,1216,1217,1221],{},[604,1218,1219],{},[610,1220,648],{},[668,1222,648],{"encoding":670},[583,1224,1226],{"className":1225,"ariaHidden":676},[675],[583,1227,1229,1233],{"className":1228},[680],[583,1230],{"className":1231,"style":1232},[684],"height:0.4306em;",[583,1234,648],{"className":1235},[689,735],":",[583,1238,1240],{"className":1239},[586],[583,1241,1243,1305],{"className":1242},[590],[583,1244,1246],{"className":1245},[594],[596,1247,1248],{"xmlns":598,"display":599},[601,1249,1250,1302],{},[604,1251,1252,1286,1288],{},[607,1253,1254,1262],{},[604,1255,1256,1258,1260],{},[610,1257,612],{},[610,1259,632],{},[610,1261,648],{},[604,1263,1264,1267,1270,1276,1278,1280,1282,1284],{},[661,1265,1266],{},"4",[619,1268,1269],{},"⋅",[607,1271,1272,1274],{},[661,1273,1038],{},[661,1275,663],{},[610,1277,632],{},[610,1279,648],{},[610,1281,617],{},[619,1283,621],{},[610,1285,624],{},[619,1287,627],{},[607,1289,1290,1292],{},[610,1291,612],{},[604,1293,1294,1296,1298,1300],{},[661,1295,663],{},[610,1297,617],{},[619,1299,621],{},[610,1301,624],{},[668,1303,1304],{"encoding":670},"\\frac{abc}{4 \\cdot \\frac{1}{2} bc \\sin \\alpha} = \\frac{a}{2 \\sin \\alpha}",[583,1306,1308,1489],{"className":1307,"ariaHidden":676},[675],[583,1309,1311,1315,1480,1483,1486],{"className":1310},[680],[583,1312],{"className":1313,"style":1314},[684],"height:2.4515em;vertical-align:-1.0801em;",[583,1316,1318,1321,1477],{"className":1317},[689],[583,1319],{"className":1320},[693,694],[583,1322,1324],{"className":1323},[607],[583,1325,1327,1468],{"className":1326},[701,702],[583,1328,1330,1465],{"className":1329},[706],[583,1331,1333,1442,1450],{"className":1332,"style":814},[710],[583,1334,1336,1339],{"style":1335},"top:-2.2649em;",[583,1337],{"className":1338,"style":719},[718],[583,1340,1342,1345,1349,1353,1356,1424,1427,1430,1433,1436,1439],{"className":1341},[689],[583,1343,1266],{"className":1344},[689],[583,1346],{"className":1347,"style":1348},[730],"margin-right:0.2222em;",[583,1350,1269],{"className":1351},[1352],"mbin",[583,1354],{"className":1355,"style":1348},[730],[583,1357,1359,1362,1421],{"className":1358},[689],[583,1360],{"className":1361},[693,694],[583,1363,1365],{"className":1364},[607],[583,1366,1368,1413],{"className":1367},[701,702],[583,1369,1371,1410],{"className":1370},[706],[583,1372,1374,1388,1396],{"className":1373,"style":1100},[710],[583,1375,1376,1379],{"style":1103},[583,1377],{"className":1378,"style":719},[718],[583,1380,1382],{"className":1381},[1110,1111,1112,1113],[583,1383,1385],{"className":1384},[689,1113],[583,1386,663],{"className":1387},[689,1113],[583,1389,1390,1393],{"style":739},[583,1391],{"className":1392,"style":719},[718],[583,1394],{"className":1395,"style":747},[746],[583,1397,1398,1401],{"style":1130},[583,1399],{"className":1400,"style":719},[718],[583,1402,1404],{"className":1403},[1110,1111,1112,1113],[583,1405,1407],{"className":1406},[689,1113],[583,1408,1038],{"className":1409},[689,1113],[583,1411,764],{"className":1412},[763],[583,1414,1416],{"className":1415},[706],[583,1417,1419],{"className":1418,"style":1152},[710],[583,1420],{},[583,1422],{"className":1423},[777,694],[583,1425,632],{"className":1426},[689,735],[583,1428,648],{"className":1429},[689,735],[583,1431],{"className":1432,"style":731},[730],[583,1434,617],{"className":1435},[726],[583,1437],{"className":1438,"style":731},[730],[583,1440,624],{"className":1441,"style":736},[689,735],[583,1443,1444,1447],{"style":739},[583,1445],{"className":1446,"style":719},[718],[583,1448],{"className":1449,"style":747},[746],[583,1451,1452,1455],{"style":750},[583,1453],{"className":1454,"style":719},[718],[583,1456,1458,1462],{"className":1457},[689],[583,1459,1461],{"className":1460},[689,735],"ab",[583,1463,648],{"className":1464},[689,735],[583,1466,764],{"className":1467},[763],[583,1469,1471],{"className":1470},[706],[583,1472,1475],{"className":1473,"style":1474},[710],"height:1.0801em;",[583,1476],{},[583,1478],{"className":1479},[777,694],[583,1481],{"className":1482,"style":781},[730],[583,1484,627],{"className":1485},[785],[583,1487],{"className":1488,"style":781},[730],[583,1490,1492,1495],{"className":1491},[680],[583,1493],{"className":1494,"style":685},[684],[583,1496,1498,1501,1566],{"className":1497},[689],[583,1499],{"className":1500},[693,694],[583,1502,1504],{"className":1503},[607],[583,1505,1507,1558],{"className":1506},[701,702],[583,1508,1510,1555],{"className":1509},[706],[583,1511,1513,1536,1544],{"className":1512,"style":711},[710],[583,1514,1515,1518],{"style":714},[583,1516],{"className":1517,"style":719},[718],[583,1519,1521,1524,1527,1530,1533],{"className":1520},[689],[583,1522,663],{"className":1523},[689],[583,1525],{"className":1526,"style":731},[730],[583,1528,617],{"className":1529},[726],[583,1531],{"className":1532,"style":731},[730],[583,1534,624],{"className":1535,"style":736},[689,735],[583,1537,1538,1541],{"style":739},[583,1539],{"className":1540,"style":719},[718],[583,1542],{"className":1543,"style":747},[746],[583,1545,1546,1549],{"style":750},[583,1547],{"className":1548,"style":719},[718],[583,1550,1552],{"className":1551},[689],[583,1553,612],{"className":1554},[689,735],[583,1556,764],{"className":1557},[763],[583,1559,1561],{"className":1560},[706],[583,1562,1564],{"className":1563,"style":771},[710],[583,1565],{},[583,1567],{"className":1568},[777,694],[558,1570,1572],{"id":1571},"beispiele","Beispiele",[563,1574,1575,1576,1606,1607,1606,1635,1664,1665,1236],{},"Rechtwinkliges Dreieck mit ",[583,1577,1579,1593],{"className":1578},[590],[583,1580,1582],{"className":1581},[594],[596,1583,1584],{"xmlns":598},[601,1585,1586,1591],{},[604,1587,1588],{},[661,1589,1590],{},"3",[668,1592,1590],{"encoding":670},[583,1594,1596],{"className":1595,"ariaHidden":676},[675],[583,1597,1599,1603],{"className":1598},[680],[583,1600],{"className":1601,"style":1602},[684],"height:0.6444em;",[583,1604,1590],{"className":1605},[689],", ",[583,1608,1610,1623],{"className":1609},[590],[583,1611,1613],{"className":1612},[594],[596,1614,1615],{"xmlns":598},[601,1616,1617,1621],{},[604,1618,1619],{},[661,1620,1266],{},[668,1622,1266],{"encoding":670},[583,1624,1626],{"className":1625,"ariaHidden":676},[675],[583,1627,1629,1632],{"className":1628},[680],[583,1630],{"className":1631,"style":1602},[684],[583,1633,1266],{"className":1634},[689],[583,1636,1638,1652],{"className":1637},[590],[583,1639,1641],{"className":1640},[594],[596,1642,1643],{"xmlns":598},[601,1644,1645,1650],{},[604,1646,1647],{},[661,1648,1649],{},"5",[668,1651,1649],{"encoding":670},[583,1653,1655],{"className":1654,"ariaHidden":676},[675],[583,1656,1658,1661],{"className":1657},[680],[583,1659],{"className":1660,"style":1602},[684],[583,1662,1649],{"className":1663},[689]," (in cm) und ",[583,1666,1668,1699],{"className":1667},[590],[583,1669,1671],{"className":1670},[594],[596,1672,1673],{"xmlns":598},[601,1674,1675,1696],{},[604,1676,1677,1679,1681,1684,1688],{},[610,1678,1031],{},[619,1680,627],{},[661,1682,1683],{},"6",[1685,1686,1687],"mtext",{}," ",[1689,1690,1691,1694],"msup",{},[1685,1692,1693],{},"cm",[661,1695,663],{},[668,1697,1698],{"encoding":670},"A = 6\\,\\text{cm}^2",[583,1700,1702,1720],{"className":1701,"ariaHidden":676},[675],[583,1703,1705,1708,1711,1714,1717],{"className":1704},[680],[583,1706],{"className":1707,"style":967},[684],[583,1709,1031],{"className":1710},[689,735],[583,1712],{"className":1713,"style":781},[730],[583,1715,627],{"className":1716},[785],[583,1718],{"className":1719,"style":781},[730],[583,1721,1723,1727,1730,1733],{"className":1722},[680],[583,1724],{"className":1725,"style":1726},[684],"height:0.8141em;",[583,1728,1683],{"className":1729},[689],[583,1731],{"className":1732,"style":731},[730],[583,1734,1736,1743],{"className":1735},[689],[583,1737,1740],{"className":1738},[689,1739],"text",[583,1741,1693],{"className":1742},[689],[583,1744,1747],{"className":1745},[1746],"msupsub",[583,1748,1750],{"className":1749},[701],[583,1751,1753],{"className":1752},[706],[583,1754,1756],{"className":1755,"style":1726},[710],[583,1757,1759,1763],{"style":1758},"top:-3.063em;margin-right:0.05em;",[583,1760],{"className":1761,"style":1762},[718],"height:2.7em;",[583,1764,1766],{"className":1765},[1110,1111,1112,1113],[583,1767,663],{"className":1768},[689,1113],[583,1770,1772],{"className":1771},[586],[583,1773,1775,1832],{"className":1774},[590],[583,1776,1778],{"className":1777},[594],[596,1779,1780],{"xmlns":598,"display":599},[601,1781,1782,1829],{},[604,1783,1784,1786,1788,1810,1812,1820,1822,1825,1827],{},[610,1785,666],{},[619,1787,627],{},[607,1789,1790,1802],{},[604,1791,1792,1794,1796,1798,1800],{},[661,1793,1590],{},[619,1795,1269],{},[661,1797,1266],{},[619,1799,1269],{},[661,1801,1649],{},[604,1803,1804,1806,1808],{},[661,1805,1266],{},[619,1807,1269],{},[661,1809,1683],{},[619,1811,627],{},[607,1813,1814,1817],{},[661,1815,1816],{},"60",[661,1818,1819],{},"24",[619,1821,627],{},[661,1823,1824],{},"2,5",[1685,1826,1687],{},[1685,1828,1693],{},[668,1830,1831],{"encoding":670},"R = \\frac{3 \\cdot 4 \\cdot 5}{4 \\cdot 6} = \\frac{60}{24} = 2{,}5\\,\\text{cm}",[583,1833,1835,1853,1968,2045],{"className":1834,"ariaHidden":676},[675],[583,1836,1838,1841,1844,1847,1850],{"className":1837},[680],[583,1839],{"className":1840,"style":967},[684],[583,1842,666],{"className":1843,"style":974},[689,735],[583,1845],{"className":1846,"style":781},[730],[583,1848,627],{"className":1849},[785],[583,1851],{"className":1852,"style":781},[730],[583,1854,1856,1860,1959,1962,1965],{"className":1855},[680],[583,1857],{"className":1858,"style":1859},[684],"height:2.0074em;vertical-align:-0.686em;",[583,1861,1863,1866,1956],{"className":1862},[689],[583,1864],{"className":1865},[693,694],[583,1867,1869],{"className":1868},[607],[583,1870,1872,1948],{"className":1871},[701,702],[583,1873,1875,1945],{"className":1874},[706],[583,1876,1879,1902,1910],{"className":1877,"style":1878},[710],"height:1.3214em;",[583,1880,1881,1884],{"style":714},[583,1882],{"className":1883,"style":719},[718],[583,1885,1887,1890,1893,1896,1899],{"className":1886},[689],[583,1888,1266],{"className":1889},[689],[583,1891],{"className":1892,"style":1348},[730],[583,1894,1269],{"className":1895},[1352],[583,1897],{"className":1898,"style":1348},[730],[583,1900,1683],{"className":1901},[689],[583,1903,1904,1907],{"style":739},[583,1905],{"className":1906,"style":719},[718],[583,1908],{"className":1909,"style":747},[746],[583,1911,1912,1915],{"style":750},[583,1913],{"className":1914,"style":719},[718],[583,1916,1918,1921,1924,1927,1930,1933,1936,1939,1942],{"className":1917},[689],[583,1919,1590],{"className":1920},[689],[583,1922],{"className":1923,"style":1348},[730],[583,1925,1269],{"className":1926},[1352],[583,1928],{"className":1929,"style":1348},[730],[583,1931,1266],{"className":1932},[689],[583,1934],{"className":1935,"style":1348},[730],[583,1937,1269],{"className":1938},[1352],[583,1940],{"className":1941,"style":1348},[730],[583,1943,1649],{"className":1944},[689],[583,1946,764],{"className":1947},[763],[583,1949,1951],{"className":1950},[706],[583,1952,1954],{"className":1953,"style":771},[710],[583,1955],{},[583,1957],{"className":1958},[777,694],[583,1960],{"className":1961,"style":781},[730],[583,1963,627],{"className":1964},[785],[583,1966],{"className":1967,"style":781},[730],[583,1969,1971,1974,2036,2039,2042],{"className":1970},[680],[583,1972],{"className":1973,"style":1859},[684],[583,1975,1977,1980,2033],{"className":1976},[689],[583,1978],{"className":1979},[693,694],[583,1981,1983],{"className":1982},[607],[583,1984,1986,2025],{"className":1985},[701,702],[583,1987,1989,2022],{"className":1988},[706],[583,1990,1992,2003,2011],{"className":1991,"style":1878},[710],[583,1993,1994,1997],{"style":714},[583,1995],{"className":1996,"style":719},[718],[583,1998,2000],{"className":1999},[689],[583,2001,1819],{"className":2002},[689],[583,2004,2005,2008],{"style":739},[583,2006],{"className":2007,"style":719},[718],[583,2009],{"className":2010,"style":747},[746],[583,2012,2013,2016],{"style":750},[583,2014],{"className":2015,"style":719},[718],[583,2017,2019],{"className":2018},[689],[583,2020,1816],{"className":2021},[689],[583,2023,764],{"className":2024},[763],[583,2026,2028],{"className":2027},[706],[583,2029,2031],{"className":2030,"style":771},[710],[583,2032],{},[583,2034],{"className":2035},[777,694],[583,2037],{"className":2038,"style":781},[730],[583,2040,627],{"className":2041},[785],[583,2043],{"className":2044,"style":781},[730],[583,2046,2048,2052,2055,2063,2066,2069],{"className":2047},[680],[583,2049],{"className":2050,"style":2051},[684],"height:0.8389em;vertical-align:-0.1944em;",[583,2053,663],{"className":2054},[689],[583,2056,2058],{"className":2057},[689],[583,2059,2062],{"className":2060},[2061],"mpunct",",",[583,2064,1649],{"className":2065},[689],[583,2067],{"className":2068,"style":731},[730],[583,2070,2072],{"className":2071},[689,1739],[583,2073,1693],{"className":2074},[689],[563,2076,2077],{},"Das ist genau die halbe Hypotenuse.",[563,2079,2080,2081,2145,2146,1236],{},"Gleichseitiges Dreieck mit ",[583,2082,2084,2106],{"className":2083},[590],[583,2085,2087],{"className":2086},[594],[596,2088,2089],{"xmlns":598},[601,2090,2091,2103],{},[604,2092,2093,2095,2097,2099,2101],{},[610,2094,612],{},[619,2096,627],{},[661,2098,1683],{},[1685,2100,1687],{},[1685,2102,1693],{},[668,2104,2105],{"encoding":670},"a = 6\\,\\text{cm}",[583,2107,2109,2127],{"className":2108,"ariaHidden":676},[675],[583,2110,2112,2115,2118,2121,2124],{"className":2111},[680],[583,2113],{"className":2114,"style":1232},[684],[583,2116,612],{"className":2117},[689,735],[583,2119],{"className":2120,"style":781},[730],[583,2122,627],{"className":2123},[785],[583,2125],{"className":2126,"style":781},[730],[583,2128,2130,2133,2136,2139],{"className":2129},[680],[583,2131],{"className":2132,"style":1602},[684],[583,2134,1683],{"className":2135},[689],[583,2137],{"className":2138,"style":731},[730],[583,2140,2142],{"className":2141},[689,1739],[583,2143,1693],{"className":2144},[689],", alle Winkel ",[583,2147,2149,2168],{"className":2148},[590],[583,2150,2152],{"className":2151},[594],[596,2153,2154],{"xmlns":598},[601,2155,2156,2165],{},[604,2157,2158],{},[1689,2159,2160,2162],{},[661,2161,1816],{},[619,2163,2164],{},"∘",[668,2166,2167],{"encoding":670},"60^\\circ",[583,2169,2171],{"className":2170,"ariaHidden":676},[675],[583,2172,2174,2178,2181],{"className":2173},[680],[583,2175],{"className":2176,"style":2177},[684],"height:0.6741em;",[583,2179,1683],{"className":2180},[689],[583,2182,2184,2188],{"className":2183},[689],[583,2185,2187],{"className":2186},[689],"0",[583,2189,2191],{"className":2190},[1746],[583,2192,2194],{"className":2193},[701],[583,2195,2197],{"className":2196},[706],[583,2198,2200],{"className":2199,"style":2177},[710],[583,2201,2202,2205],{"style":1758},[583,2203],{"className":2204,"style":1762},[718],[583,2206,2208],{"className":2207},[1110,1111,1112,1113],[583,2209,2164],{"className":2210},[1352,1113],[583,2212,2214],{"className":2213},[586],[583,2215,2217,2280],{"className":2216},[590],[583,2218,2220],{"className":2219},[594],[596,2221,2222],{"xmlns":598,"display":599},[601,2223,2224,2277],{},[604,2225,2226,2228,2230,2248,2250,2259,2261,2263,2267,2270,2273,2275],{},[610,2227,666],{},[619,2229,627],{},[607,2231,2232,2234],{},[661,2233,1683],{},[604,2235,2236,2238,2240,2242],{},[661,2237,663],{},[610,2239,617],{},[619,2241,621],{},[1689,2243,2244,2246],{},[661,2245,1816],{},[619,2247,2164],{},[619,2249,627],{},[607,2251,2252,2254],{},[661,2253,1683],{},[2255,2256,2257],"msqrt",{},[661,2258,1590],{},[619,2260,627],{},[661,2262,663],{},[2255,2264,2265],{},[661,2266,1590],{},[619,2268,2269],{},"≈",[661,2271,2272],{},"3,46",[1685,2274,1687],{},[1685,2276,1693],{},[668,2278,2279],{"encoding":670},"R = \\frac{6}{2 \\sin 60^\\circ} = \\frac{6}{\\sqrt{3}} = 2\\sqrt{3} \\approx 3{,}46\\,\\text{cm}",[583,2281,2283,2301,2421,2562,2631],{"className":2282,"ariaHidden":676},[675],[583,2284,2286,2289,2292,2295,2298],{"className":2285},[680],[583,2287],{"className":2288,"style":967},[684],[583,2290,666],{"className":2291,"style":974},[689,735],[583,2293],{"className":2294,"style":781},[730],[583,2296,627],{"className":2297},[785],[583,2299],{"className":2300,"style":781},[730],[583,2302,2304,2307,2412,2415,2418],{"className":2303},[680],[583,2305],{"className":2306,"style":1859},[684],[583,2308,2310,2313,2409],{"className":2309},[689],[583,2311],{"className":2312},[693,694],[583,2314,2316],{"className":2315},[607],[583,2317,2319,2401],{"className":2318},[701,702],[583,2320,2322,2398],{"className":2321},[706],[583,2323,2325,2379,2387],{"className":2324,"style":1878},[710],[583,2326,2327,2330],{"style":714},[583,2328],{"className":2329,"style":719},[718],[583,2331,2333,2336,2339,2342,2345,2348],{"className":2332},[689],[583,2334,663],{"className":2335},[689],[583,2337],{"className":2338,"style":731},[730],[583,2340,617],{"className":2341},[726],[583,2343],{"className":2344,"style":731},[730],[583,2346,1683],{"className":2347},[689],[583,2349,2351,2354],{"className":2350},[689],[583,2352,2187],{"className":2353},[689],[583,2355,2357],{"className":2356},[1746],[583,2358,2360],{"className":2359},[701],[583,2361,2363],{"className":2362},[706],[583,2364,2367],{"className":2365,"style":2366},[710],"height:0.6001em;",[583,2368,2370,2373],{"style":2369},"top:-2.989em;margin-right:0.05em;",[583,2371],{"className":2372,"style":1762},[718],[583,2374,2376],{"className":2375},[1110,1111,1112,1113],[583,2377,2164],{"className":2378},[1352,1113],[583,2380,2381,2384],{"style":739},[583,2382],{"className":2383,"style":719},[718],[583,2385],{"className":2386,"style":747},[746],[583,2388,2389,2392],{"style":750},[583,2390],{"className":2391,"style":719},[718],[583,2393,2395],{"className":2394},[689],[583,2396,1683],{"className":2397},[689],[583,2399,764],{"className":2400},[763],[583,2402,2404],{"className":2403},[706],[583,2405,2407],{"className":2406,"style":771},[710],[583,2408],{},[583,2410],{"className":2411},[777,694],[583,2413],{"className":2414,"style":781},[730],[583,2416,627],{"className":2417},[785],[583,2419],{"className":2420,"style":781},[730],[583,2422,2424,2428,2553,2556,2559],{"className":2423},[680],[583,2425],{"className":2426,"style":2427},[684],"height:2.2514em;vertical-align:-0.93em;",[583,2429,2431,2434,2550],{"className":2430},[689],[583,2432],{"className":2433},[693,694],[583,2435,2437],{"className":2436},[607],[583,2438,2440,2541],{"className":2439},[701,702],[583,2441,2443,2538],{"className":2442},[706],[583,2444,2446,2519,2527],{"className":2445,"style":1878},[710],[583,2447,2449,2452],{"style":2448},"top:-2.2028em;",[583,2450],{"className":2451,"style":719},[718],[583,2453,2455],{"className":2454},[689],[583,2456,2459],{"className":2457},[689,2458],"sqrt",[583,2460,2462,2510],{"className":2461},[701,702],[583,2463,2465,2507],{"className":2464},[706],[583,2466,2469,2484],{"className":2467,"style":2468},[710],"height:0.9072em;",[583,2470,2474,2477],{"className":2471,"style":2473},[2472],"svg-align","top:-3em;",[583,2475],{"className":2476,"style":719},[718],[583,2478,2481],{"className":2479,"style":2480},[689],"padding-left:0.833em;",[583,2482,1590],{"className":2483},[689],[583,2485,2487,2490],{"style":2486},"top:-2.8672em;",[583,2488],{"className":2489,"style":719},[718],[583,2491,2495],{"className":2492,"style":2494},[2493],"hide-tail","min-width:0.853em;height:1.08em;",[2496,2497,2503],"svg",{"xmlns":2498,"width":2499,"height":2500,"viewBox":2501,"preserveAspectRatio":2502},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.08em","0 0 400000 1080","xMinYMin slice",[2504,2505],"path",{"d":2506},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[583,2508,764],{"className":2509},[763],[583,2511,2513],{"className":2512},[706],[583,2514,2517],{"className":2515,"style":2516},[710],"height:0.1328em;",[583,2518],{},[583,2520,2521,2524],{"style":739},[583,2522],{"className":2523,"style":719},[718],[583,2525],{"className":2526,"style":747},[746],[583,2528,2529,2532],{"style":750},[583,2530],{"className":2531,"style":719},[718],[583,2533,2535],{"className":2534},[689],[583,2536,1683],{"className":2537},[689],[583,2539,764],{"className":2540},[763],[583,2542,2544],{"className":2543},[706],[583,2545,2548],{"className":2546,"style":2547},[710],"height:0.93em;",[583,2549],{},[583,2551],{"className":2552},[777,694],[583,2554],{"className":2555,"style":781},[730],[583,2557,627],{"className":2558},[785],[583,2560],{"className":2561,"style":781},[730],[583,2563,2565,2569,2572,2622,2625,2628],{"className":2564},[680],[583,2566],{"className":2567,"style":2568},[684],"height:1.04em;vertical-align:-0.0839em;",[583,2570,663],{"className":2571},[689],[583,2573,2575],{"className":2574},[689,2458],[583,2576,2578,2613],{"className":2577},[701,702],[583,2579,2581,2610],{"className":2580},[706],[583,2582,2585,2597],{"className":2583,"style":2584},[710],"height:0.9561em;",[583,2586,2588,2591],{"className":2587,"style":2473},[2472],[583,2589],{"className":2590,"style":719},[718],[583,2592,2594],{"className":2593,"style":2480},[689],[583,2595,1590],{"className":2596},[689],[583,2598,2600,2603],{"style":2599},"top:-2.9161em;",[583,2601],{"className":2602,"style":719},[718],[583,2604,2606],{"className":2605,"style":2494},[2493],[2496,2607,2608],{"xmlns":2498,"width":2499,"height":2500,"viewBox":2501,"preserveAspectRatio":2502},[2504,2609],{"d":2506},[583,2611,764],{"className":2612},[763],[583,2614,2616],{"className":2615},[706],[583,2617,2620],{"className":2618,"style":2619},[710],"height:0.0839em;",[583,2621],{},[583,2623],{"className":2624,"style":781},[730],[583,2626,2269],{"className":2627},[785],[583,2629],{"className":2630,"style":781},[730],[583,2632,2634,2637,2640,2646,2650,2653],{"className":2633},[680],[583,2635],{"className":2636,"style":2051},[684],[583,2638,1590],{"className":2639},[689],[583,2641,2643],{"className":2642},[689],[583,2644,2062],{"className":2645},[2061],[583,2647,2649],{"className":2648},[689],"46",[583,2651],{"className":2652,"style":731},[730],[583,2654,2656],{"className":2655},[689,1739],[583,2657,1693],{"className":2658},[689],[563,2660,2661,2662,1606,2690,1606,2718,1664,2746,2841,2842,3010],{},"Dreieck mit ",[583,2663,2665,2678],{"className":2664},[590],[583,2666,2668],{"className":2667},[594],[596,2669,2670],{"xmlns":598},[601,2671,2672,2676],{},[604,2673,2674],{},[661,2675,1649],{},[668,2677,1649],{"encoding":670},[583,2679,2681],{"className":2680,"ariaHidden":676},[675],[583,2682,2684,2687],{"className":2683},[680],[583,2685],{"className":2686,"style":1602},[684],[583,2688,1649],{"className":2689},[689],[583,2691,2693,2706],{"className":2692},[590],[583,2694,2696],{"className":2695},[594],[596,2697,2698],{"xmlns":598},[601,2699,2700,2704],{},[604,2701,2702],{},[661,2703,1649],{},[668,2705,1649],{"encoding":670},[583,2707,2709],{"className":2708,"ariaHidden":676},[675],[583,2710,2712,2715],{"className":2711},[680],[583,2713],{"className":2714,"style":1602},[684],[583,2716,1649],{"className":2717},[689],[583,2719,2721,2734],{"className":2720},[590],[583,2722,2724],{"className":2723},[594],[596,2725,2726],{"xmlns":598},[601,2727,2728,2732],{},[604,2729,2730],{},[661,2731,1683],{},[668,2733,1683],{"encoding":670},[583,2735,2737],{"className":2736,"ariaHidden":676},[675],[583,2738,2740,2743],{"className":2739},[680],[583,2741],{"className":2742,"style":1602},[684],[583,2744,1683],{"className":2745},[689],[583,2747,2749,2776],{"className":2748},[590],[583,2750,2752],{"className":2751},[594],[596,2753,2754],{"xmlns":598},[601,2755,2756,2773],{},[604,2757,2758,2760,2762,2765,2767],{},[610,2759,1031],{},[619,2761,627],{},[661,2763,2764],{},"12",[1685,2766,1687],{},[1689,2768,2769,2771],{},[1685,2770,1693],{},[661,2772,663],{},[668,2774,2775],{"encoding":670},"A = 12\\,\\text{cm}^2",[583,2777,2779,2797],{"className":2778,"ariaHidden":676},[675],[583,2780,2782,2785,2788,2791,2794],{"className":2781},[680],[583,2783],{"className":2784,"style":967},[684],[583,2786,1031],{"className":2787},[689,735],[583,2789],{"className":2790,"style":781},[730],[583,2792,627],{"className":2793},[785],[583,2795],{"className":2796,"style":781},[730],[583,2798,2800,2803,2806,2809],{"className":2799},[680],[583,2801],{"className":2802,"style":1726},[684],[583,2804,2764],{"className":2805},[689],[583,2807],{"className":2808,"style":731},[730],[583,2810,2812,2818],{"className":2811},[689],[583,2813,2815],{"className":2814},[689,1739],[583,2816,1693],{"className":2817},[689],[583,2819,2821],{"className":2820},[1746],[583,2822,2824],{"className":2823},[701],[583,2825,2827],{"className":2826},[706],[583,2828,2830],{"className":2829,"style":1726},[710],[583,2831,2832,2835],{"style":1758},[583,2833],{"className":2834,"style":1762},[718],[583,2836,2838],{"className":2837},[1110,1111,1112,1113],[583,2839,663],{"className":2840},[689,1113],": ",[583,2843,2845,2878],{"className":2844},[590],[583,2846,2848],{"className":2847},[594],[596,2849,2850],{"xmlns":598},[601,2851,2852,2875],{},[604,2853,2854,2856,2858,2866,2868,2871,2873],{},[610,2855,666],{},[619,2857,627],{},[607,2859,2860,2863],{},[661,2861,2862],{},"150",[661,2864,2865],{},"48",[619,2867,627],{},[661,2869,2870],{},"3,125",[1685,2872,1687],{},[1685,2874,1693],{},[668,2876,2877],{"encoding":670},"R = \\frac{150}{48} = 3{,}125\\,\\text{cm}",[583,2879,2881,2899,2982],{"className":2880,"ariaHidden":676},[675],[583,2882,2884,2887,2890,2893,2896],{"className":2883},[680],[583,2885],{"className":2886,"style":967},[684],[583,2888,666],{"className":2889,"style":974},[689,735],[583,2891],{"className":2892,"style":781},[730],[583,2894,627],{"className":2895},[785],[583,2897],{"className":2898,"style":781},[730],[583,2900,2902,2905,2973,2976,2979],{"className":2901},[680],[583,2903],{"className":2904,"style":1081},[684],[583,2906,2908,2911,2970],{"className":2907},[689],[583,2909],{"className":2910},[693,694],[583,2912,2914],{"className":2913},[607],[583,2915,2917,2962],{"className":2916},[701,702],[583,2918,2920,2959],{"className":2919},[706],[583,2921,2923,2937,2945],{"className":2922,"style":1100},[710],[583,2924,2925,2928],{"style":1103},[583,2926],{"className":2927,"style":719},[718],[583,2929,2931],{"className":2930},[1110,1111,1112,1113],[583,2932,2934],{"className":2933},[689,1113],[583,2935,2865],{"className":2936},[689,1113],[583,2938,2939,2942],{"style":739},[583,2940],{"className":2941,"style":719},[718],[583,2943],{"className":2944,"style":747},[746],[583,2946,2947,2950],{"style":1130},[583,2948],{"className":2949,"style":719},[718],[583,2951,2953],{"className":2952},[1110,1111,1112,1113],[583,2954,2956],{"className":2955},[689,1113],[583,2957,2862],{"className":2958},[689,1113],[583,2960,764],{"className":2961},[763],[583,2963,2965],{"className":2964},[706],[583,2966,2968],{"className":2967,"style":1152},[710],[583,2969],{},[583,2971],{"className":2972},[777,694],[583,2974],{"className":2975,"style":781},[730],[583,2977,627],{"className":2978},[785],[583,2980],{"className":2981,"style":781},[730],[583,2983,2985,2988,2991,2997,3001,3004],{"className":2984},[680],[583,2986],{"className":2987,"style":2051},[684],[583,2989,1590],{"className":2990},[689],[583,2992,2994],{"className":2993},[689],[583,2995,2062],{"className":2996},[2061],[583,2998,3000],{"className":2999},[689],"125",[583,3002],{"className":3003,"style":731},[730],[583,3005,3007],{"className":3006},[689,1739],[583,3008,1693],{"className":3009},[689],".",[558,3012,3014],{"id":3013},"typische-fehler","Typische Fehler",[3016,3017,3018,3022,3095],"ul",{},[3019,3020,3021],"li",{},"Einen Winkel nehmen, der nicht gegenüber der verwendeten Seite liegt.",[3019,3023,3024,3025,3059,3060,3094],{},"Durch ",[583,3026,3028,3044],{"className":3027},[590],[583,3029,3031],{"className":3030},[594],[596,3032,3033],{"xmlns":598},[601,3034,3035,3041],{},[604,3036,3037,3039],{},[661,3038,663],{},[610,3040,1031],{},[668,3042,3043],{"encoding":670},"2A",[583,3045,3047],{"className":3046,"ariaHidden":676},[675],[583,3048,3050,3053,3056],{"className":3049},[680],[583,3051],{"className":3052,"style":967},[684],[583,3054,663],{"className":3055},[689],[583,3057,1031],{"className":3058},[689,735]," statt durch ",[583,3061,3063,3079],{"className":3062},[590],[583,3064,3066],{"className":3065},[594],[596,3067,3068],{"xmlns":598},[601,3069,3070,3076],{},[604,3071,3072,3074],{},[661,3073,1266],{},[610,3075,1031],{},[668,3077,3078],{"encoding":670},"4A",[583,3080,3082],{"className":3081,"ariaHidden":676},[675],[583,3083,3085,3088,3091],{"className":3084},[680],[583,3086],{"className":3087,"style":967},[684],[583,3089,1266],{"className":3090},[689],[583,3092,1031],{"className":3093},[689,735]," teilen.",[3019,3096,3097],{},"Den Umkreismittelpunkt über die Winkelhalbierenden suchen. Das liefert den Inkreismittelpunkt.",{"title":3099,"searchDepth":3100,"depth":3100,"links":3101},"",2,[3102,3103,3104,3105],{"id":560,"depth":3100,"text":561},{"id":1012,"depth":3100,"text":1013},{"id":1571,"depth":3100,"text":1572},{"id":3013,"depth":3100,"text":3014},[3107,3110,3113,3115,3118,3121],{"lhs":3108,"rhs":3109},"a*b*c\u002F(4*(1\u002F2*b*c*sin(w)))","a\u002F(2*sin(w))",{"lhs":3111,"rhs":3112},"3*4*5\u002F(4*6)","2.5",{"lhs":3114,"rhs":3112},"5\u002F(2*sin(90*pi\u002F180))",{"lhs":3116,"rhs":3117},"6\u002F(2*sin(60*pi\u002F180))","2*sqrt(3)",{"lhs":3119,"rhs":3120},"round(2*sqrt(3), 2)","3.46",{"lhs":3122,"rhs":3123},"5*5*6\u002F(4*12)","3.125","0^\\circ \u003C \\alpha \u003C 180^\\circ","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmn>0\u003C\u002Fmn>\u003Cmo>∘\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo><\u003C\u002Fmo>\u003Cmi>α\u003C\u002Fmi>\u003Cmo><\u003C\u002Fmo>\u003Cmsup>\u003Cmn>180\u003C\u002Fmn>\u003Cmo>∘\u003C\u002Fmo>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">0^\\circ < \\alpha < 180^\\circ\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7132em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6741em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mbin mtight\">∘\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\"><\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\"><\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6741em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">18\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6741em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mbin mtight\">∘\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md",{"caption":3129,"x":3130,"y":3133,"items":3134},"Dreieck mit den Seiten a, b, c und dem Winkel α, dazu der Umkreis mit dem Mittelpunkt M und dem Radius R",[3131,3132],-2.5,2.5,[3131,3132],[3135,3139,3151,3156,3161,3168,3173,3177,3181,3184,3189,3193],{"type":3136,"center":3137,"r":3100},"circle",[3138,3138],0,{"type":3140,"points":3141,"fill":3150},"polygon",[3142,3145,3147],[3143,3144],-1.732,-1,[3146,3144],1.732,[3148,3149],-0.347,1.97,true,{"type":3152,"from":3153,"to":3154,"style":3155},"segment",[3138,3138],[3146,3144],"dashed",{"type":3157,"at":3158,"from":3159,"to":3160,"label":624},"angle",[3143,3144],[3146,3144],[3148,3149],{"type":3162,"at":3163,"label":3164,"offset":3165},"point",[3138,3138],"M",[3166,3167],-8,-12,{"type":3169,"at":3170,"text":666},"label",[3171,3172],0.62,-0.08,{"type":3169,"at":3174,"text":612},[3175,3176],0.98,0.69,{"type":3169,"at":3178,"text":632},[3179,3180],-1.31,0.61,{"type":3169,"at":3182,"text":648},[3138,3183],-1.32,{"type":3162,"at":3185,"label":1031,"offset":3186},[3143,3144],[3187,3188],-18,8,{"type":3162,"at":3190,"label":3191,"offset":3192},[3146,3144],"B",[3188,3188],{"type":3162,"at":3194,"label":3195,"offset":3196},[3148,3149],"C",[3197,3198],-4,-14,"\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"0 0 240 240\" width=\"240\" height=\"240\" role=\"img\" aria-label=\"Dreieck mit den Seiten a, b, c und dem Winkel α, dazu der Umkreis mit dem Mittelpunkt M und dem Radius R\" class=\"formula-figure-svg\">\u003Cdefs>\u003Cmarker id=\"fig-circumcircle-radius-arrow\" viewBox=\"0 0 10 10\" refX=\"9\" refY=\"5\" markerWidth=\"7\" markerHeight=\"7\" orient=\"auto-start-reverse\">\u003Cpath d=\"M 0 0 L 10 5 L 0 10 z\" class=\"fig-arrowhead\"\u002F>\u003C\u002Fmarker>\u003CclipPath id=\"fig-circumcircle-radius-clip\">\u003Crect x=\"0\" y=\"0\" width=\"240\" height=\"240\"\u002F>\u003C\u002FclipPath>\u003C\u002Fdefs>\u003Ccircle cx=\"120\" cy=\"120\" r=\"96\" class=\"fig-main\"\u002F>\u003Cpolygon points=\"36.86,168 203.14,168 103.34,25.44\" class=\"fig-main fig-fill\"\u002F>\u003Cline x1=\"120\" y1=\"120\" x2=\"203.14\" y2=\"168\" class=\"fig-dashed\"\u002F>\u003Cpath d=\"M 36.86 168 L 58.86 168 A 22 22 0 0 0 46.16 148.06 Z\" class=\"fig-angle\"\u002F>\u003Ctext x=\"65.54\" y=\"149.73\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">α\u003C\u002Ftext>\u003Ccircle cx=\"120\" cy=\"120\" r=\"3\" class=\"fig-dot\"\u002F>\u003Ctext x=\"112\" y=\"108\" text-anchor=\"start\" dominant-baseline=\"middle\" class=\"fig-label fig-upright\">M\u003C\u002Ftext>\u003Ctext x=\"149.76\" y=\"123.84\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">R\u003C\u002Ftext>\u003Ctext x=\"167.04\" y=\"86.88\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">a\u003C\u002Ftext>\u003Ctext x=\"57.12\" y=\"90.72\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">b\u003C\u002Ftext>\u003Ctext x=\"120\" y=\"183.36\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">c\u003C\u002Ftext>\u003Ccircle cx=\"36.86\" cy=\"168\" r=\"3\" class=\"fig-dot\"\u002F>\u003Ctext x=\"18.86\" y=\"176\" text-anchor=\"start\" dominant-baseline=\"middle\" class=\"fig-label fig-upright\">A\u003C\u002Ftext>\u003Ccircle cx=\"203.14\" cy=\"168\" r=\"3\" class=\"fig-dot\"\u002F>\u003Ctext x=\"211.14\" y=\"176\" text-anchor=\"start\" dominant-baseline=\"middle\" class=\"fig-label fig-upright\">B\u003C\u002Ftext>\u003Ccircle cx=\"103.34\" cy=\"25.44\" r=\"3\" class=\"fig-dot\"\u002F>\u003Ctext x=\"99.34\" y=\"11.440000000000001\" text-anchor=\"start\" dominant-baseline=\"middle\" class=\"fig-label fig-upright\">C\u003C\u002Ftext>\u003C\u002Fsvg>","circumcircle-radius","R = \\frac{a}{2 \\sin \\alpha} = \\frac{a \\cdot b \\cdot c}{4A}","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmi>R\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmfrac>\u003Cmi>a\u003C\u002Fmi>\u003Cmrow>\u003Cmn>2\u003C\u002Fmn>\u003Cmi>sin\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>α\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmfrac>\u003Cmo>=\u003C\u002Fmo>\u003Cmfrac>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmi>b\u003C\u002Fmi>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmi>c\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmrow>\u003Cmn>4\u003C\u002Fmn>\u003Cmi>A\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmfrac>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">R = \\frac{a}{2 \\sin \\alpha} = \\frac{a \\cdot b \\cdot c}{4A}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0077em;\">R\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.7936em;vertical-align:-0.686em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.1076em;\">\u003Cspan style=\"top:-2.314em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">2\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">sin\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.677em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.686em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:2.0574em;vertical-align:-0.686em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.3714em;\">\u003Cspan style=\"top:-2.314em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">4\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.677em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.686em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","upper",{},"\u002Fde\u002Fformulas\u002Fmath\u002Fgeometry\u002Fumkreisradius",[],[3208,3209,3210,3211],"law-of-sines","incircle-radius","thales-theorem","heron-formula",{"title":548,"description":3099},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fumkreisradius","Der Umkreisradius folgt aus einer Seite und dem gegenüberliegenden Winkel oder aus allen drei Seiten und der Fläche.",[3216,3217,3218,3219,3220],"umkreis","dreieck","radius","mittelsenkrechte","sinussatz",[3222,3225,3229,3233],{"symbol":666,"meaning":3223,"symbolHtml":3224},"Radius des Umkreises","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>R\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">R\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0077em;\">R\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":3226,"meaning":3227,"symbolHtml":3228},"a, b, c","Seitenlängen des Dreiecks","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>b\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>c\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a, b, c\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":3230,"meaning":3231,"symbolHtml":3232},"\\alpha","Winkel gegenüber der Seite a","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>α\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\alpha\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":1031,"meaning":3234,"symbolHtml":3235},"Flächeninhalt des Dreiecks","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>A\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">A\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","r1J5YUfFm9DnKz0dwsAlX_lCJp3JI_J-3vI9DsC1CuA",[3238,3241,3244,3247],{"stem":3239,"title":3240,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Finkreisradius","Inkreisradius eines Dreiecks",{"stem":3242,"title":3243,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fsatz-des-heron","Satz des Heron",{"stem":3245,"title":3246,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fsatz-des-thales","Satz des Thales",{"stem":3248,"title":3249,"category":518},"de\u002Fformulas\u002Fmath\u002Ftrigonometry\u002Fsinussatz","Sinussatz",1793824270526]