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Über dem einen Bogen betrachtet man zwei Arten von Winkeln:",[572,573,574,630],"ul",{},[575,576,577,578,581,582,629],"li",{},"den ",[567,579,580],{},"Mittelpunktswinkel"," ",[583,584,587,610],"span",{"className":585},[586],"katex",[583,588,591],{"className":589},[590],"katex-mathml",[592,593,595],"math",{"xmlns":594},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[596,597,598,605],"semantics",{},[599,600,601],"mrow",{},[602,603,604],"mi",{},"μ",[606,607,609],"annotation",{"encoding":608},"application\u002Fx-tex","\\mu",[583,611,615],{"className":612,"ariaHidden":614},[613],"katex-html","true",[583,616,619,624],{"className":617},[618],"base",[583,620],{"className":621,"style":623},[622],"strut","height:0.625em;vertical-align:-0.1944em;",[583,625,604],{"className":626},[627,628],"mord","mathnormal"," mit dem Scheitel im Kreismittelpunkt,",[575,631,632,633,581,636,668],{},"die ",[567,634,635],{},"Umfangswinkel",[583,637,639,654],{"className":638},[586],[583,640,642],{"className":641},[590],[592,643,644],{"xmlns":594},[596,645,646,651],{},[599,647,648],{},[602,649,650],{},"α",[606,652,653],{"encoding":608},"\\alpha",[583,655,657],{"className":656,"ariaHidden":614},[613],[583,658,660,664],{"className":659},[618],[583,661],{"className":662,"style":663},[622],"height:0.4306em;",[583,665,650],{"className":666,"style":667},[627,628],"margin-right:0.0037em;"," mit dem Scheitel irgendwo auf dem anderen Bogen.",[563,670,671],{},"Der Satz sagt zweierlei:",[673,674,675,682],"ol",{},[575,676,677,678,681],{},"Alle Umfangswinkel über demselben Bogen sind ",[567,679,680],{},"gleich groß",", egal wo der Scheitel auf dem gegenüberliegenden Bogen liegt.",[575,683,684,685,688],{},"Jeder Umfangswinkel ist ",[567,686,687],{},"halb so groß"," wie der Mittelpunktswinkel.",[563,690,691,692,720,721,829,830,833,834,896],{},"Liegt der Scheitel dagegen auf dem Bogen, über dem ",[583,693,695,708],{"className":694},[586],[583,696,698],{"className":697},[590],[592,699,700],{"xmlns":594},[596,701,702,706],{},[599,703,704],{},[602,705,604],{},[606,707,609],{"encoding":608},[583,709,711],{"className":710,"ariaHidden":614},[613],[583,712,714,717],{"className":713},[618],[583,715],{"className":716,"style":623},[622],[583,718,604],{"className":719},[627,628]," steht, ist der Umfangswinkel ",[583,722,724,752],{"className":723},[586],[583,725,727],{"className":726},[590],[592,728,729],{"xmlns":594},[596,730,731,749],{},[599,732,733,744,747],{},[734,735,736,740],"msup",{},[737,738,739],"mn",{},"180",[741,742,743],"mo",{},"∘",[741,745,746],{},"−",[602,748,650],{},[606,750,751],{"encoding":608},"180^\\circ - \\alpha",[583,753,755,820],{"className":754,"ariaHidden":614},[613],[583,756,758,762,766,809,814,817],{"className":757},[618],[583,759],{"className":760,"style":761},[622],"height:0.7574em;vertical-align:-0.0833em;",[583,763,765],{"className":764},[627],"18",[583,767,769,773],{"className":768},[627],[583,770,772],{"className":771},[627],"0",[583,774,777],{"className":775},[776],"msupsub",[583,778,781],{"className":779},[780],"vlist-t",[583,782,785],{"className":783},[784],"vlist-r",[583,786,790],{"className":787,"style":789},[788],"vlist","height:0.6741em;",[583,791,793,798],{"style":792},"top:-3.063em;margin-right:0.05em;",[583,794],{"className":795,"style":797},[796],"pstrut","height:2.7em;",[583,799,805],{"className":800},[801,802,803,804],"sizing","reset-size6","size3","mtight",[583,806,743],{"className":807},[808,804],"mbin",[583,810],{"className":811,"style":813},[812],"mspace","margin-right:0.2222em;",[583,815,746],{"className":816},[808],[583,818],{"className":819,"style":813},[812],[583,821,823,826],{"className":822},[618],[583,824],{"className":825,"style":663},[622],[583,827,650],{"className":828,"style":667},[627,628],". Deshalb ergänzen sich gegenüberliegende Winkel in einem ",[567,831,832],{},"Sehnenviereck"," zu ",[583,835,837,855],{"className":836},[586],[583,838,840],{"className":839},[590],[592,841,842],{"xmlns":594},[596,843,844,852],{},[599,845,846],{},[734,847,848,850],{},[737,849,739],{},[741,851,743],{},[606,853,854],{"encoding":608},"180^\\circ",[583,856,858],{"className":857,"ariaHidden":614},[613],[583,859,861,864,867],{"className":860},[618],[583,862],{"className":863,"style":789},[622],[583,865,765],{"className":866},[627],[583,868,870,873],{"className":869},[627],[583,871,772],{"className":872},[627],[583,874,876],{"className":875},[776],[583,877,879],{"className":878},[780],[583,880,882],{"className":881},[784],[583,883,885],{"className":884,"style":789},[788],[583,886,887,890],{"style":792},[583,888],{"className":889,"style":797},[796],[583,891,893],{"className":892},[801,802,803,804],[583,894,743],{"className":895},[808,804],".",[563,898,899,900,903,904,991,992,896],{},"Der ",[567,901,902],{},"Satz des Thales"," ist der Sonderfall ",[583,905,907,930],{"className":906},[586],[583,908,910],{"className":909},[590],[592,911,912],{"xmlns":594},[596,913,914,927],{},[599,915,916,918,921],{},[602,917,604],{},[741,919,920],{},"=",[734,922,923,925],{},[737,924,739],{},[741,926,743],{},[606,928,929],{"encoding":608},"\\mu = 180^\\circ",[583,931,933,953],{"className":932,"ariaHidden":614},[613],[583,934,936,939,942,946,950],{"className":935},[618],[583,937],{"className":938,"style":623},[622],[583,940,604],{"className":941},[627,628],[583,943],{"className":944,"style":945},[812],"margin-right:0.2778em;",[583,947,920],{"className":948},[949],"mrel",[583,951],{"className":952,"style":945},[812],[583,954,956,959,962],{"className":955},[618],[583,957],{"className":958,"style":789},[622],[583,960,765],{"className":961},[627],[583,963,965,968],{"className":964},[627],[583,966,772],{"className":967},[627],[583,969,971],{"className":970},[776],[583,972,974],{"className":973},[780],[583,975,977],{"className":976},[784],[583,978,980],{"className":979,"style":789},[788],[583,981,982,985],{"style":792},[583,983],{"className":984,"style":797},[796],[583,986,988],{"className":987},[801,802,803,804],[583,989,743],{"className":990},[808,804],": Ist die Sehne ein Durchmesser, sind alle Umfangswinkel ",[583,993,995,1014],{"className":994},[586],[583,996,998],{"className":997},[590],[592,999,1000],{"xmlns":594},[596,1001,1002,1011],{},[599,1003,1004],{},[734,1005,1006,1009],{},[737,1007,1008],{},"90",[741,1010,743],{},[606,1012,1013],{"encoding":608},"90^\\circ",[583,1015,1017],{"className":1016,"ariaHidden":614},[613],[583,1018,1020,1023,1027],{"className":1019},[618],[583,1021],{"className":1022,"style":789},[622],[583,1024,1026],{"className":1025},[627],"9",[583,1028,1030,1033],{"className":1029},[627],[583,1031,772],{"className":1032},[627],[583,1034,1036],{"className":1035},[776],[583,1037,1039],{"className":1038},[780],[583,1040,1042],{"className":1041},[784],[583,1043,1045],{"className":1044,"style":789},[788],[583,1046,1047,1050],{"style":792},[583,1048],{"className":1049,"style":797},[796],[583,1051,1053],{"className":1052},[801,802,803,804],[583,1054,743],{"className":1055},[808,804],[558,1057,1059],{"id":1058},"beispiele","Beispiele",[563,1061,1062,1063,1149],{},"Der Mittelpunktswinkel über einer Sehne ist ",[583,1064,1066,1089],{"className":1065},[586],[583,1067,1069],{"className":1068},[590],[592,1070,1071],{"xmlns":594},[596,1072,1073,1086],{},[599,1074,1075,1077,1079],{},[602,1076,604],{},[741,1078,920],{},[734,1080,1081,1084],{},[737,1082,1083],{},"110",[741,1085,743],{},[606,1087,1088],{"encoding":608},"\\mu = 110^\\circ",[583,1090,1092,1110],{"className":1091,"ariaHidden":614},[613],[583,1093,1095,1098,1101,1104,1107],{"className":1094},[618],[583,1096],{"className":1097,"style":623},[622],[583,1099,604],{"className":1100},[627,628],[583,1102],{"className":1103,"style":945},[812],[583,1105,920],{"className":1106},[949],[583,1108],{"className":1109,"style":945},[812],[583,1111,1113,1116,1120],{"className":1112},[618],[583,1114],{"className":1115,"style":789},[622],[583,1117,1119],{"className":1118},[627],"11",[583,1121,1123,1126],{"className":1122},[627],[583,1124,772],{"className":1125},[627],[583,1127,1129],{"className":1128},[776],[583,1130,1132],{"className":1131},[780],[583,1133,1135],{"className":1134},[784],[583,1136,1138],{"className":1137,"style":789},[788],[583,1139,1140,1143],{"style":792},[583,1141],{"className":1142,"style":797},[796],[583,1144,1146],{"className":1145},[801,802,803,804],[583,1147,743],{"className":1148},[808,804],". Jeder Umfangswinkel auf dem gegenüberliegenden Bogen ist",[583,1151,1154],{"className":1152},[1153],"katex-display",[583,1155,1157,1195],{"className":1156},[586],[583,1158,1160],{"className":1159},[590],[592,1161,1163],{"xmlns":594,"display":1162},"block",[596,1164,1165,1192],{},[599,1166,1167,1169,1171,1183,1185],{},[602,1168,650],{},[741,1170,920],{},[1172,1173,1174,1180],"mfrac",{},[734,1175,1176,1178],{},[737,1177,1083],{},[741,1179,743],{},[737,1181,1182],{},"2",[741,1184,920],{},[734,1186,1187,1190],{},[737,1188,1189],{},"55",[741,1191,743],{},[606,1193,1194],{"encoding":608},"\\alpha = \\frac{110^\\circ}{2} = 55^\\circ",[583,1196,1198,1216,1337],{"className":1197,"ariaHidden":614},[613],[583,1199,1201,1204,1207,1210,1213],{"className":1200},[618],[583,1202],{"className":1203,"style":663},[622],[583,1205,650],{"className":1206,"style":667},[627,628],[583,1208],{"className":1209,"style":945},[812],[583,1211,920],{"className":1212},[949],[583,1214],{"className":1215,"style":945},[812],[583,1217,1219,1223,1328,1331,1334],{"className":1218},[618],[583,1220],{"className":1221,"style":1222},[622],"height:2.0371em;vertical-align:-0.686em;",[583,1224,1226,1231,1324],{"className":1225},[627],[583,1227],{"className":1228},[1229,1230],"mopen","nulldelimiter",[583,1232,1234],{"className":1233},[1172],[583,1235,1238,1315],{"className":1236},[780,1237],"vlist-t2",[583,1239,1241,1310],{"className":1240},[784],[583,1242,1245,1258,1269],{"className":1243,"style":1244},[788],"height:1.3511em;",[583,1246,1248,1252],{"style":1247},"top:-2.314em;",[583,1249],{"className":1250,"style":1251},[796],"height:3em;",[583,1253,1255],{"className":1254},[627],[583,1256,1182],{"className":1257},[627],[583,1259,1261,1264],{"style":1260},"top:-3.23em;",[583,1262],{"className":1263,"style":1251},[796],[583,1265],{"className":1266,"style":1268},[1267],"frac-line","border-bottom-width:0.04em;",[583,1270,1272,1275],{"style":1271},"top:-3.677em;",[583,1273],{"className":1274,"style":1251},[796],[583,1276,1278,1281],{"className":1277},[627],[583,1279,1119],{"className":1280},[627],[583,1282,1284,1287],{"className":1283},[627],[583,1285,772],{"className":1286},[627],[583,1288,1290],{"className":1289},[776],[583,1291,1293],{"className":1292},[780],[583,1294,1296],{"className":1295},[784],[583,1297,1299],{"className":1298,"style":789},[788],[583,1300,1301,1304],{"style":792},[583,1302],{"className":1303,"style":797},[796],[583,1305,1307],{"className":1306},[801,802,803,804],[583,1308,743],{"className":1309},[808,804],[583,1311,1314],{"className":1312},[1313],"vlist-s","​",[583,1316,1318],{"className":1317},[784],[583,1319,1322],{"className":1320,"style":1321},[788],"height:0.686em;",[583,1323],{},[583,1325],{"className":1326},[1327,1230],"mclose",[583,1329],{"className":1330,"style":945},[812],[583,1332,920],{"className":1333},[949],[583,1335],{"className":1336,"style":945},[812],[583,1338,1340,1344,1348],{"className":1339},[618],[583,1341],{"className":1342,"style":1343},[622],"height:0.7241em;",[583,1345,1347],{"className":1346},[627],"5",[583,1349,1351,1354],{"className":1350},[627],[583,1352,1347],{"className":1353},[627],[583,1355,1357],{"className":1356},[776],[583,1358,1360],{"className":1359},[780],[583,1361,1363],{"className":1362},[784],[583,1364,1366],{"className":1365,"style":1343},[788],[583,1367,1369,1372],{"style":1368},"top:-3.113em;margin-right:0.05em;",[583,1370],{"className":1371,"style":797},[796],[583,1373,1375],{"className":1374},[801,802,803,804],[583,1376,743],{"className":1377},[808,804],[563,1379,1380,1381,896],{},"Ein Umfangswinkel auf der anderen Seite der Sehne ist ",[583,1382,1384,1419],{"className":1383},[586],[583,1385,1387],{"className":1386},[590],[592,1388,1389],{"xmlns":594},[596,1390,1391,1416],{},[599,1392,1393,1399,1401,1407,1409],{},[734,1394,1395,1397],{},[737,1396,739],{},[741,1398,743],{},[741,1400,746],{},[734,1402,1403,1405],{},[737,1404,1189],{},[741,1406,743],{},[741,1408,920],{},[734,1410,1411,1414],{},[737,1412,1413],{},"125",[741,1415,743],{},[606,1417,1418],{"encoding":608},"180^\\circ - 55^\\circ = 125^\\circ",[583,1420,1422,1469,1516],{"className":1421,"ariaHidden":614},[613],[583,1423,1425,1428,1431,1460,1463,1466],{"className":1424},[618],[583,1426],{"className":1427,"style":761},[622],[583,1429,765],{"className":1430},[627],[583,1432,1434,1437],{"className":1433},[627],[583,1435,772],{"className":1436},[627],[583,1438,1440],{"className":1439},[776],[583,1441,1443],{"className":1442},[780],[583,1444,1446],{"className":1445},[784],[583,1447,1449],{"className":1448,"style":789},[788],[583,1450,1451,1454],{"style":792},[583,1452],{"className":1453,"style":797},[796],[583,1455,1457],{"className":1456},[801,802,803,804],[583,1458,743],{"className":1459},[808,804],[583,1461],{"className":1462,"style":813},[812],[583,1464,746],{"className":1465},[808],[583,1467],{"className":1468,"style":813},[812],[583,1470,1472,1475,1478,1507,1510,1513],{"className":1471},[618],[583,1473],{"className":1474,"style":789},[622],[583,1476,1347],{"className":1477},[627],[583,1479,1481,1484],{"className":1480},[627],[583,1482,1347],{"className":1483},[627],[583,1485,1487],{"className":1486},[776],[583,1488,1490],{"className":1489},[780],[583,1491,1493],{"className":1492},[784],[583,1494,1496],{"className":1495,"style":789},[788],[583,1497,1498,1501],{"style":792},[583,1499],{"className":1500,"style":797},[796],[583,1502,1504],{"className":1503},[801,802,803,804],[583,1505,743],{"className":1506},[808,804],[583,1508],{"className":1509,"style":945},[812],[583,1511,920],{"className":1512},[949],[583,1514],{"className":1515,"style":945},[812],[583,1517,1519,1522,1526],{"className":1518},[618],[583,1520],{"className":1521,"style":789},[622],[583,1523,1525],{"className":1524},[627],"12",[583,1527,1529,1532],{"className":1528},[627],[583,1530,1347],{"className":1531},[627],[583,1533,1535],{"className":1534},[776],[583,1536,1538],{"className":1537},[780],[583,1539,1541],{"className":1540},[784],[583,1542,1544],{"className":1543,"style":789},[788],[583,1545,1546,1549],{"style":792},[583,1547],{"className":1548,"style":797},[796],[583,1550,1552],{"className":1551},[801,802,803,804],[583,1553,743],{"className":1554},[808,804],[563,1556,1557,1558,1641,1642,896],{},"Ist die Sehne ein Durchmesser, gilt ",[583,1559,1561,1582],{"className":1560},[586],[583,1562,1564],{"className":1563},[590],[592,1565,1566],{"xmlns":594},[596,1567,1568,1580],{},[599,1569,1570,1572,1574],{},[602,1571,604],{},[741,1573,920],{},[734,1575,1576,1578],{},[737,1577,739],{},[741,1579,743],{},[606,1581,929],{"encoding":608},[583,1583,1585,1603],{"className":1584,"ariaHidden":614},[613],[583,1586,1588,1591,1594,1597,1600],{"className":1587},[618],[583,1589],{"className":1590,"style":623},[622],[583,1592,604],{"className":1593},[627,628],[583,1595],{"className":1596,"style":945},[812],[583,1598,920],{"className":1599},[949],[583,1601],{"className":1602,"style":945},[812],[583,1604,1606,1609,1612],{"className":1605},[618],[583,1607],{"className":1608,"style":789},[622],[583,1610,765],{"className":1611},[627],[583,1613,1615,1618],{"className":1614},[627],[583,1616,772],{"className":1617},[627],[583,1619,1621],{"className":1620},[776],[583,1622,1624],{"className":1623},[780],[583,1625,1627],{"className":1626},[784],[583,1628,1630],{"className":1629,"style":789},[788],[583,1631,1632,1635],{"style":792},[583,1633],{"className":1634,"style":797},[796],[583,1636,1638],{"className":1637},[801,802,803,804],[583,1639,743],{"className":1640},[808,804]," und ",[583,1643,1645,1667],{"className":1644},[586],[583,1646,1648],{"className":1647},[590],[592,1649,1650],{"xmlns":594},[596,1651,1652,1664],{},[599,1653,1654,1656,1658],{},[602,1655,650],{},[741,1657,920],{},[734,1659,1660,1662],{},[737,1661,1008],{},[741,1663,743],{},[606,1665,1666],{"encoding":608},"\\alpha = 90^\\circ",[583,1668,1670,1688],{"className":1669,"ariaHidden":614},[613],[583,1671,1673,1676,1679,1682,1685],{"className":1672},[618],[583,1674],{"className":1675,"style":663},[622],[583,1677,650],{"className":1678,"style":667},[627,628],[583,1680],{"className":1681,"style":945},[812],[583,1683,920],{"className":1684},[949],[583,1686],{"className":1687,"style":945},[812],[583,1689,1691,1694,1697],{"className":1690},[618],[583,1692],{"className":1693,"style":789},[622],[583,1695,1026],{"className":1696},[627],[583,1698,1700,1703],{"className":1699},[627],[583,1701,772],{"className":1702},[627],[583,1704,1706],{"className":1705},[776],[583,1707,1709],{"className":1708},[780],[583,1710,1712],{"className":1711},[784],[583,1713,1715],{"className":1714,"style":789},[788],[583,1716,1717,1720],{"style":792},[583,1718],{"className":1719,"style":797},[796],[583,1721,1723],{"className":1722},[801,802,803,804],[583,1724,743],{"className":1725},[808,804],[558,1727,1729],{"id":1728},"typische-fehler","Typische Fehler",[572,1731,1732,1735,1738],{},[575,1733,1734],{},"Verdoppeln statt halbieren: Der Mittelpunktswinkel ist der größere.",[575,1736,1737],{},"Umfangswinkel von verschiedenen Seiten der Sehne gleichsetzen. Gleich groß sind nur die auf demselben Bogen.",[575,1739,1740],{},"Einen Winkel als Umfangswinkel ansehen, dessen Scheitel nicht auf dem Kreis liegt.",{"title":1742,"searchDepth":1743,"depth":1743,"links":1744},"",2,[1745,1746,1747],{"id":560,"depth":1743,"text":561},{"id":1058,"depth":1743,"text":1059},{"id":1728,"depth":1743,"text":1729},[1749,1752,1754,1756,1758],{"lhs":1750,"rhs":1751},"acos(dot([1 - cos(4), 0 - sin(4)], [cos(1.2) - cos(4), sin(1.2) - sin(4)]) \u002F (norm([1 - cos(4), 0 - sin(4)]) * norm([cos(1.2) - cos(4), sin(1.2) - sin(4)])))","1.2\u002F2",{"lhs":1753,"rhs":1751},"acos(dot([1 - cos(5.5), 0 - sin(5.5)], [cos(1.2) - cos(5.5), sin(1.2) - sin(5.5)]) \u002F (norm([1 - cos(5.5), 0 - sin(5.5)]) * norm([cos(1.2) - cos(5.5), sin(1.2) - sin(5.5)])))",{"lhs":1755,"rhs":1189},"110\u002F2",{"lhs":1757,"rhs":1413},"180 - 55",{"lhs":1759,"rhs":1008},"180\u002F2","\\text{Scheitel von } \\alpha \\text{ auf dem Kreisbogen, der } \\mu \\text{ gegenüberliegt}","\u003Cspan class=\"katex\">Scheitel von\u003C\u002Fspan> \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> \u003Cspan class=\"katex\">auf dem Kreisbogen, der\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>μ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mu\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:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">gegenüberliegt\u003C\u002Fspan>",null,"md",{"caption":1765,"x":1766,"y":1769,"items":1770},"Umfangswinkel α und Mittelpunktswinkel μ über derselben Sehne",[1767,1768],-2.6,2.6,[1767,1768],[1771,1777,1785,1790,1796,1799,1802,1805,1808,1811,1813,1815],{"type":1772,"center":1773,"r":1775,"style":1776},"circle",[1774,1774],0,2.2,"muted",{"type":1778,"from":1779,"to":1782,"style":1784},"segment",[1780,1781],-1.802,-1.262,[1783,1781],1.802,"dashed",{"type":1786,"at":1787,"from":1788,"to":1789,"label":604},"angle",[1774,1774],[1780,1781],[1783,1781],{"type":1786,"at":1791,"from":1794,"to":1795,"label":650},[1792,1793],-0.569,2.125,[1780,1781],[1783,1781],{"type":1778,"from":1797,"to":1798},[1774,1774],[1780,1781],{"type":1778,"from":1800,"to":1801},[1774,1774],[1783,1781],{"type":1778,"from":1803,"to":1804},[1792,1793],[1780,1781],{"type":1778,"from":1806,"to":1807},[1792,1793],[1783,1781],{"type":1809,"at":1810},"point",[1780,1781],{"type":1809,"at":1812},[1783,1781],{"type":1809,"at":1814},[1792,1793],{"type":1809,"at":1816,"label":1817,"offset":1818},[1774,1774],"M",[1819,1820],-6,-14,"\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"0 0 249.6 249.6\" width=\"249.6\" height=\"249.6\" role=\"img\" aria-label=\"Umfangswinkel α und Mittelpunktswinkel μ über derselben Sehne\" class=\"formula-figure-svg\">\u003Cdefs>\u003Cmarker id=\"fig-inscribed-angle-theorem-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-inscribed-angle-theorem-clip\">\u003Crect x=\"0\" y=\"0\" width=\"249.6\" height=\"249.6\"\u002F>\u003C\u002FclipPath>\u003C\u002Fdefs>\u003Ccircle cx=\"124.8\" cy=\"124.8\" r=\"105.6\" class=\"fig-muted\"\u002F>\u003Cline x1=\"38.3\" y1=\"185.38\" x2=\"211.3\" y2=\"185.38\" class=\"fig-dashed\"\u002F>\u003Cpath d=\"M 124.8 124.8 L 106.78 137.42 A 22 22 0 0 0 142.82 137.42 Z\" class=\"fig-angle\"\u002F>\u003Ctext x=\"124.8\" y=\"158.8\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">μ\u003C\u002Ftext>\u003Cpath d=\"M 97.49 22.8 L 89.96 43.47 A 22 22 0 0 0 110.11 40.82 Z\" class=\"fig-angle\"\u002F>\u003Ctext x=\"101.92\" y=\"56.51\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">α\u003C\u002Ftext>\u003Cline x1=\"124.8\" y1=\"124.8\" x2=\"38.3\" y2=\"185.38\" class=\"fig-main\"\u002F>\u003Cline x1=\"124.8\" y1=\"124.8\" x2=\"211.3\" y2=\"185.38\" class=\"fig-main\"\u002F>\u003Cline x1=\"97.49\" y1=\"22.8\" x2=\"38.3\" y2=\"185.38\" class=\"fig-main\"\u002F>\u003Cline x1=\"97.49\" y1=\"22.8\" x2=\"211.3\" y2=\"185.38\" class=\"fig-main\"\u002F>\u003Ccircle cx=\"38.3\" cy=\"185.38\" r=\"3\" class=\"fig-dot\"\u002F>\u003Ccircle cx=\"211.3\" cy=\"185.38\" r=\"3\" class=\"fig-dot\"\u002F>\u003Ccircle cx=\"97.49\" cy=\"22.8\" r=\"3\" class=\"fig-dot\"\u002F>\u003Ccircle cx=\"124.8\" cy=\"124.8\" r=\"3\" class=\"fig-dot\"\u002F>\u003Ctext x=\"118.8\" y=\"110.8\" text-anchor=\"start\" dominant-baseline=\"middle\" class=\"fig-label fig-upright\">M\u003C\u002Ftext>\u003C\u002Fsvg>","inscribed-angle-theorem","\\alpha = \\frac{\\mu}{2}","\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>α\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmfrac>\u003Cmi>μ\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmfrac>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\alpha = \\frac{\\mu}{2}\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>\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>\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\">μ\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>","middle",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Fgeometry\u002Fumfangswinkelsatz",[],[1831,1832,1833],"thales-theorem","triangle-angle-sum","arc-length",{"title":548,"description":1742},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fumfangswinkelsatz","Jeder Umfangswinkel über einem Kreisbogen ist halb so groß wie der Mittelpunktswinkel über diesem 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