1917:
31:
1904:
355:
41:
350:{\displaystyle {\begin{array}{cl}\implies &\angle PG_{2}T=\angle PTG_{1}\\\implies &\triangle PTG_{2}\sim \triangle PG_{1}T\\\implies &{\frac {|PT|}{|PG_{2}|}}={\frac {|PG_{1}|}{|PT|}}\\\implies &|PT|^{2}=|PG_{1}|\cdot |PG_{2}|\end{array}}}
531:
673:
1777:
1855:
546:, the tangent-secant theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle, namely, the
1158:
694:
666:
1702:
1432:
603:
1397:
1765:
1377:
1168:
1832:
659:
444:
586:
568:
1941:
1801:
1734:
1367:
1247:
1870:
1627:
1515:
690:
543:
1510:
539:
1579:
1206:
1022:
997:
1741:
1712:
1072:
927:
1875:
1152:
1609:
1837:
1813:
1687:
1622:
1563:
1500:
1490:
1226:
1145:
1007:
917:
797:
1860:
1808:
1707:
1535:
1485:
1470:
1465:
1236:
1037:
972:
962:
912:
1908:
1760:
1545:
1422:
1345:
1293:
1095:
1002:
852:
1916:
1604:
1885:
1825:
1789:
1632:
1455:
1402:
1372:
1362:
1271:
1134:
1027:
942:
897:
877:
722:
707:
389:
46:
1865:
1784:
1772:
1753:
1717:
1637:
1555:
1540:
1530:
1480:
1475:
1417:
1286:
1178:
1107:
1042:
1032:
932:
902:
842:
817:
742:
732:
717:
1921:
1880:
1820:
1748:
1614:
1407:
1382:
1350:
947:
892:
857:
752:
361:
1589:
1188:
1594:
1505:
1355:
1281:
1254:
1017:
837:
827:
762:
682:
634:
599:
590:
582:
572:
564:
1796:
1667:
1584:
1340:
1328:
1276:
1012:
547:
1437:
1427:
1321:
1047:
35:
1697:
1692:
1520:
1412:
1392:
1220:
777:
747:
624:
17:
1935:
1657:
1525:
1495:
1316:
1124:
1067:
1599:
1387:
1062:
822:
812:
369:
30:
1213:
1101:
807:
792:
535:
The tangent-secant theorem can be proven using similar triangles (see graphic).
373:
616:
1199:
1118:
1057:
1052:
992:
977:
922:
907:
862:
802:
787:
767:
737:
702:
598:. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008,
952:
937:
887:
782:
772:
757:
727:
651:
642:
27:
Geometry theorem relating line segments created by a secant and tangent line
637:
1089:
867:
712:
1662:
987:
982:
882:
872:
847:
377:
957:
832:
385:
381:
1333:
1311:
967:
29:
655:
384:. This result is found as Proposition 36 in Book 3 of
447:
44:
1848:
1726:
1676:
1650:
1572:
1554:
1453:
1446:
1302:
1264:
1081:
689:
525:
349:
526:{\displaystyle |PT|^{2}=|PG_{1}|\cdot |PG_{2}|}
667:
8:
561:The VNR Concise Encyclopedia of Mathematics
1680:
1450:
674:
660:
652:
265:
261:
157:
153:
105:
101:
53:
49:
518:
512:
500:
492:
486:
474:
465:
460:
448:
446:
338:
332:
320:
312:
306:
294:
285:
280:
268:
249:
238:
231:
225:
213:
210:
199:
193:
181:
174:
163:
160:
140:
121:
91:
66:
45:
43:
1703:Latin translations of the 12th century
1433:Straightedge and compass construction
7:
1398:Incircle and excircles of a triangle
401:intersecting the circle at points
130:
108:
78:
56:
25:
423:intersecting the circle at point
1915:
1902:
439:, the following equation holds:
1735:A History of Greek Mathematics
1248:The Quadrature of the Parabola
519:
501:
493:
475:
461:
449:
339:
321:
313:
295:
281:
269:
262:
250:
239:
232:
214:
200:
182:
175:
164:
154:
102:
50:
1:
1516:Intersecting secants theorem
544:intersecting secants theorem
1511:Intersecting chords theorem
1378:Doctrine of proportionality
606:, pp. 415-417 (German)
596:Schülerduden - Mathematik I
540:intersecting chords theorem
1958:
1207:On the Sphere and Cylinder
1160:On the Sizes and Distances
368:describes the relation of
1909:Ancient Greece portal
1898:
1713:Philosophy of mathematics
1683:
1628:Ptolemy's table of chords
683:Ancient Greek mathematics
380:line with the associated
36:alternate segment theorem
1580:Aristarchus's inequality
1153:On Conoids and Spheroids
626:Power of a Point Theorem
1688:Ancient Greek astronomy
1501:Inscribed angle theorem
1491:Greek geometric algebra
1146:Measurement of a Circle
579:Revolutions in Geometry
1942:Theorems about circles
1922:Mathematics portal
1708:Non-Euclidean geometry
1663:Mouseion of Alexandria
1536:Tangent-secant theorem
1486:Geometric mean theorem
1471:Exterior angle theorem
1466:Angle bisector theorem
1170:On Sizes and Distances
618:Tangent Secant Theorem
548:power of point theorem
527:
366:tangent-secant theorem
357:
351:
18:Tangent-secant theorem
1610:Pappus's area theorem
1546:Theorem of the gnomon
1423:Quadratrix of Hippias
1346:Circles of Apollonius
1294:Problem of Apollonius
1272:Constructible numbers
1096:Archimedes Palimpsest
528:
352:
33:
1826:prehistoric counting
1623:Ptolemy's inequality
1564:Apollonius's theorem
1403:Method of exhaustion
1373:Diophantine equation
1363:Circumscribed circle
1180:On the Moving Sphere
629:auf cut-the-knot.org
577:Michael L. O'Leary:
445:
42:
1912: •
1718:Neusis construction
1638:Spiral of Theodorus
1531:Pythagorean theorem
1476:Euclidean algorithm
1418:Lune of Hippocrates
1287:Squaring the circle
1043:Theon of Alexandria
718:Aristaeus the Elder
435:intersect at point
34:Beginning with the
1605:Menelaus's theorem
1595:Irrational numbers
1408:Parallel postulate
1383:Euclidean geometry
1351:Apollonian circles
893:Isidore of Miletus
635:Weisstein, Eric W.
563:. Springer, 2012,
523:
362:Euclidean geometry
358:
347:
345:
1929:
1928:
1894:
1893:
1646:
1645:
1633:Ptolemy's theorem
1506:Intercept theorem
1356:Apollonian gasket
1282:Doubling the cube
1255:The Sand Reckoner
604:978-3-411-04208-1
255:
205:
16:(Redirected from
1949:
1920:
1919:
1907:
1906:
1905:
1681:
1668:Platonic Academy
1615:Problem II.8 of
1585:Crossbar theorem
1541:Thales's theorem
1481:Euclid's theorem
1451:
1368:Commensurability
1329:Axiomatic system
1277:Angle trisection
1242:
1232:
1194:
1184:
1174:
1164:
1140:
1130:
1113:
676:
669:
662:
653:
648:
647:
621:at proofwiki.org
532:
530:
529:
524:
522:
517:
516:
504:
496:
491:
490:
478:
470:
469:
464:
452:
438:
434:
430:
426:
422:
418:
409:
400:
356:
354:
353:
348:
346:
342:
337:
336:
324:
316:
311:
310:
298:
290:
289:
284:
272:
256:
254:
253:
242:
236:
235:
230:
229:
217:
211:
206:
204:
203:
198:
197:
185:
179:
178:
167:
161:
145:
144:
126:
125:
96:
95:
71:
70:
21:
1957:
1956:
1952:
1951:
1950:
1948:
1947:
1946:
1932:
1931:
1930:
1925:
1914:
1903:
1901:
1890:
1856:Arabian/Islamic
1844:
1833:numeral systems
1722:
1672:
1642:
1590:Heron's formula
1568:
1550:
1442:
1438:Triangle center
1428:Regular polygon
1305:and definitions
1304:
1298:
1260:
1240:
1230:
1192:
1182:
1172:
1162:
1138:
1128:
1111:
1077:
1048:Theon of Smyrna
693:
685:
680:
633:
632:
613:
581:. Wiley, 2010,
556:
508:
482:
459:
443:
442:
436:
432:
428:
427:and given that
424:
420:
417:
411:
408:
402:
398:
397:Given a secant
344:
343:
328:
302:
279:
266:
258:
257:
237:
221:
212:
189:
180:
162:
158:
150:
149:
136:
117:
106:
98:
97:
87:
62:
54:
40:
39:
28:
23:
22:
15:
12:
11:
5:
1955:
1953:
1945:
1944:
1934:
1933:
1927:
1926:
1899:
1896:
1895:
1892:
1891:
1889:
1888:
1883:
1878:
1873:
1868:
1863:
1858:
1852:
1850:
1849:Other cultures
1846:
1845:
1843:
1842:
1841:
1840:
1830:
1829:
1828:
1818:
1817:
1816:
1806:
1805:
1804:
1794:
1793:
1792:
1782:
1781:
1780:
1770:
1769:
1768:
1758:
1757:
1756:
1746:
1745:
1744:
1730:
1728:
1724:
1723:
1721:
1720:
1715:
1710:
1705:
1700:
1698:Greek numerals
1695:
1693:Attic numerals
1690:
1684:
1678:
1674:
1673:
1671:
1670:
1665:
1660:
1654:
1652:
1648:
1647:
1644:
1643:
1641:
1640:
1635:
1630:
1625:
1620:
1612:
1607:
1602:
1597:
1592:
1587:
1582:
1576:
1574:
1570:
1569:
1567:
1566:
1560:
1558:
1552:
1551:
1549:
1548:
1543:
1538:
1533:
1528:
1523:
1521:Law of cosines
1518:
1513:
1508:
1503:
1498:
1493:
1488:
1483:
1478:
1473:
1468:
1462:
1460:
1448:
1444:
1443:
1441:
1440:
1435:
1430:
1425:
1420:
1415:
1413:Platonic solid
1410:
1405:
1400:
1395:
1393:Greek numerals
1390:
1385:
1380:
1375:
1370:
1365:
1360:
1359:
1358:
1353:
1343:
1338:
1337:
1336:
1326:
1325:
1324:
1319:
1308:
1306:
1300:
1299:
1297:
1296:
1291:
1290:
1289:
1284:
1279:
1268:
1266:
1262:
1261:
1259:
1258:
1251:
1244:
1234:
1224:
1221:Planisphaerium
1217:
1210:
1203:
1196:
1186:
1176:
1166:
1156:
1149:
1142:
1132:
1122:
1115:
1105:
1098:
1093:
1085:
1083:
1079:
1078:
1076:
1075:
1070:
1065:
1060:
1055:
1050:
1045:
1040:
1035:
1030:
1025:
1020:
1015:
1010:
1005:
1000:
995:
990:
985:
980:
975:
970:
965:
960:
955:
950:
945:
940:
935:
930:
925:
920:
915:
910:
905:
900:
895:
890:
885:
880:
875:
870:
865:
860:
855:
850:
845:
840:
835:
830:
825:
820:
815:
810:
805:
800:
795:
790:
785:
780:
775:
770:
765:
760:
755:
750:
745:
740:
735:
730:
725:
720:
715:
710:
705:
699:
697:
691:Mathematicians
687:
686:
681:
679:
678:
671:
664:
656:
650:
649:
630:
622:
612:
611:External links
609:
608:
607:
593:
575:
555:
552:
521:
515:
511:
507:
503:
499:
495:
489:
485:
481:
477:
473:
468:
463:
458:
455:
451:
419:and a tangent
415:
406:
341:
335:
331:
327:
323:
319:
315:
309:
305:
301:
297:
293:
288:
283:
278:
275:
271:
267:
264:
260:
259:
252:
248:
245:
241:
234:
228:
224:
220:
216:
209:
202:
196:
192:
188:
184:
177:
173:
170:
166:
159:
156:
152:
151:
148:
143:
139:
135:
132:
129:
124:
120:
116:
113:
110:
107:
104:
100:
99:
94:
90:
86:
83:
80:
77:
74:
69:
65:
61:
58:
55:
52:
48:
47:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1954:
1943:
1940:
1939:
1937:
1924:
1923:
1918:
1911:
1910:
1897:
1887:
1884:
1882:
1879:
1877:
1874:
1872:
1869:
1867:
1864:
1862:
1859:
1857:
1854:
1853:
1851:
1847:
1839:
1836:
1835:
1834:
1831:
1827:
1824:
1823:
1822:
1819:
1815:
1812:
1811:
1810:
1807:
1803:
1800:
1799:
1798:
1795:
1791:
1788:
1787:
1786:
1783:
1779:
1776:
1775:
1774:
1771:
1767:
1764:
1763:
1762:
1759:
1755:
1752:
1751:
1750:
1747:
1743:
1739:
1738:
1737:
1736:
1732:
1731:
1729:
1725:
1719:
1716:
1714:
1711:
1709:
1706:
1704:
1701:
1699:
1696:
1694:
1691:
1689:
1686:
1685:
1682:
1679:
1675:
1669:
1666:
1664:
1661:
1659:
1656:
1655:
1653:
1649:
1639:
1636:
1634:
1631:
1629:
1626:
1624:
1621:
1619:
1618:
1613:
1611:
1608:
1606:
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1586:
1583:
1581:
1578:
1577:
1575:
1571:
1565:
1562:
1561:
1559:
1557:
1553:
1547:
1544:
1542:
1539:
1537:
1534:
1532:
1529:
1527:
1526:Pons asinorum
1524:
1522:
1519:
1517:
1514:
1512:
1509:
1507:
1504:
1502:
1499:
1497:
1496:Hinge theorem
1494:
1492:
1489:
1487:
1484:
1482:
1479:
1477:
1474:
1472:
1469:
1467:
1464:
1463:
1461:
1459:
1458:
1452:
1449:
1445:
1439:
1436:
1434:
1431:
1429:
1426:
1424:
1421:
1419:
1416:
1414:
1411:
1409:
1406:
1404:
1401:
1399:
1396:
1394:
1391:
1389:
1386:
1384:
1381:
1379:
1376:
1374:
1371:
1369:
1366:
1364:
1361:
1357:
1354:
1352:
1349:
1348:
1347:
1344:
1342:
1339:
1335:
1332:
1331:
1330:
1327:
1323:
1320:
1318:
1315:
1314:
1313:
1310:
1309:
1307:
1301:
1295:
1292:
1288:
1285:
1283:
1280:
1278:
1275:
1274:
1273:
1270:
1269:
1267:
1263:
1257:
1256:
1252:
1250:
1249:
1245:
1243:
1239:
1235:
1233:
1229:
1225:
1223:
1222:
1218:
1216:
1215:
1211:
1209:
1208:
1204:
1202:
1201:
1197:
1195:
1191:
1187:
1185:
1181:
1177:
1175:
1171:
1167:
1165:
1163:(Aristarchus)
1161:
1157:
1155:
1154:
1150:
1148:
1147:
1143:
1141:
1137:
1133:
1131:
1127:
1123:
1121:
1120:
1116:
1114:
1110:
1106:
1104:
1103:
1099:
1097:
1094:
1092:
1091:
1087:
1086:
1084:
1080:
1074:
1071:
1069:
1068:Zeno of Sidon
1066:
1064:
1061:
1059:
1056:
1054:
1051:
1049:
1046:
1044:
1041:
1039:
1036:
1034:
1031:
1029:
1026:
1024:
1021:
1019:
1016:
1014:
1011:
1009:
1006:
1004:
1001:
999:
996:
994:
991:
989:
986:
984:
981:
979:
976:
974:
971:
969:
966:
964:
961:
959:
956:
954:
951:
949:
946:
944:
941:
939:
936:
934:
931:
929:
926:
924:
921:
919:
916:
914:
911:
909:
906:
904:
901:
899:
896:
894:
891:
889:
886:
884:
881:
879:
876:
874:
871:
869:
866:
864:
861:
859:
856:
854:
851:
849:
846:
844:
841:
839:
836:
834:
831:
829:
826:
824:
821:
819:
816:
814:
811:
809:
806:
804:
801:
799:
796:
794:
791:
789:
786:
784:
781:
779:
776:
774:
771:
769:
766:
764:
761:
759:
756:
754:
751:
749:
746:
744:
741:
739:
736:
734:
731:
729:
726:
724:
721:
719:
716:
714:
711:
709:
706:
704:
701:
700:
698:
696:
692:
688:
684:
677:
672:
670:
665:
663:
658:
657:
654:
645:
644:
639:
636:
631:
628:
627:
623:
620:
619:
615:
614:
610:
605:
601:
597:
594:
592:
588:
587:9780470591796
584:
580:
576:
574:
570:
569:9789401169820
566:
562:
559:S. Gottwald:
558:
557:
553:
551:
549:
545:
541:
536:
533:
513:
509:
505:
497:
487:
483:
479:
471:
466:
456:
453:
440:
414:
405:
395:
393:
392:
387:
383:
379:
375:
372:created by a
371:
370:line segments
367:
363:
333:
329:
325:
317:
307:
303:
299:
291:
286:
276:
273:
246:
243:
226:
222:
218:
207:
194:
190:
186:
171:
168:
146:
141:
137:
133:
127:
122:
118:
114:
111:
92:
88:
84:
81:
75:
72:
67:
63:
59:
37:
32:
19:
1913:
1900:
1742:Thomas Heath
1733:
1616:
1600:Law of sines
1456:
1388:Golden ratio
1253:
1246:
1237:
1231:(Theodosius)
1227:
1219:
1212:
1205:
1198:
1189:
1179:
1173:(Hipparchus)
1169:
1159:
1151:
1144:
1135:
1125:
1117:
1112:(Apollonius)
1108:
1100:
1088:
1063:Zeno of Elea
823:Eratosthenes
813:Dionysodorus
641:
625:
617:
595:
578:
560:
537:
534:
441:
412:
403:
396:
390:
365:
359:
1809:mathematics
1617:Arithmetica
1214:Ostomachion
1183:(Autolycus)
1102:Arithmetica
878:Hippocrates
808:Dinostratus
793:Dicaearchus
723:Aristarchus
1861:Babylonian
1761:arithmetic
1727:History of
1556:Apollonius
1241:(Menelaus)
1200:On Spirals
1119:Catoptrics
1058:Xenocrates
1053:Thymaridas
1038:Theodosius
1023:Theaetetus
1003:Simplicius
993:Pythagoras
978:Posidonius
963:Philonides
923:Nicomachus
918:Metrodorus
908:Menaechmus
863:Hipparchus
853:Heliodorus
803:Diophantus
788:Democritus
768:Chrysippus
738:Archimedes
733:Apollonius
703:Anaxagoras
695:(timeline)
554:References
1322:Inscribed
1082:Treatises
1073:Zenodorus
1033:Theodorus
1008:Sosigenes
953:Philolaus
938:Oenopides
933:Nicoteles
928:Nicomedes
888:Hypsicles
783:Ctesibius
773:Cleomedes
758:Callippus
743:Autolycus
728:Aristotle
708:Anthemius
643:MathWorld
538:Like the
498:⋅
318:⋅
263:⟹
155:⟹
131:△
128:∼
109:△
103:⟹
79:∠
57:∠
51:⟹
1936:Category
1886:Japanese
1871:Egyptian
1814:timeline
1802:timeline
1790:timeline
1785:geometry
1778:timeline
1773:calculus
1766:timeline
1754:timeline
1457:Elements
1303:Concepts
1265:Problems
1238:Spherics
1228:Spherics
1193:(Euclid)
1139:(Euclid)
1136:Elements
1129:(Euclid)
1090:Almagest
998:Serenus
973:Porphyry
913:Menelaus
868:Hippasus
843:Eutocius
818:Domninus
713:Archytas
542:and the
391:Elements
1866:Chinese
1821:numbers
1749:algebra
1677:Related
1651:Centers
1447:Results
1317:Central
988:Ptolemy
983:Proclus
948:Perseus
903:Marinus
883:Hypatia
873:Hippias
848:Geminus
838:Eudoxus
828:Eudemus
798:Diocles
638:"Chord"
573:175-176
378:tangent
1881:Indian
1658:Cyrene
1190:Optics
1109:Conics
1028:Theano
1018:Thales
1013:Sporus
958:Philon
943:Pappus
833:Euclid
763:Carpus
753:Bryson
602:
589:, p.
585:
571:, pp.
567:
386:Euclid
382:circle
376:and a
374:secant
364:, the
1876:Incan
1797:logic
1573:Other
1341:Chord
1334:Axiom
1312:Angle
968:Plato
858:Heron
778:Conon
1838:list
1126:Data
898:Leon
748:Bion
600:ISBN
583:ISBN
565:ISBN
431:and
410:and
1740:by
1454:In
591:161
394:.
388:'s
360:In
38:,
1938::
640:.
550:.
675:e
668:t
661:v
646:.
520:|
514:2
510:G
506:P
502:|
494:|
488:1
484:G
480:P
476:|
472:=
467:2
462:|
457:T
454:P
450:|
437:P
433:t
429:g
425:T
421:t
416:2
413:G
407:1
404:G
399:g
340:|
334:2
330:G
326:P
322:|
314:|
308:1
304:G
300:P
296:|
292:=
287:2
282:|
277:T
274:P
270:|
251:|
247:T
244:P
240:|
233:|
227:1
223:G
219:P
215:|
208:=
201:|
195:2
191:G
187:P
183:|
176:|
172:T
169:P
165:|
147:T
142:1
138:G
134:P
123:2
119:G
115:T
112:P
93:1
89:G
85:T
82:P
76:=
73:T
68:2
64:G
60:P
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
