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48:
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1025:
transforms the plane in a way that maps circles into circles, and pencils of circles into pencils of circles. The type of the pencil is preserved: the inversion of an elliptic pencil is another elliptic pencil, the inversion of a hyperbolic pencil is another hyperbolic pencil, and the inversion of a
1099:
More generally, for every pencil of circles there exists a unique pencil consisting of the circles that are perpendicular to the first pencil. If one pencil is elliptic, its perpendicular pencil is hyperbolic, and vice versa; in this case the two pencils form a set of
Apollonian circles. The pencil
661:
809:
235:
1108:
Apollonian trajectories have been shown to be followed in their motion by vortex cores or other defined pseudospin states in some physical systems involving interferential or coupled fields, such photonic or coupled
475:
311:
605:
1100:
of circles perpendicular to a parabolic pencil is also parabolic; it consists of the circles that have the same common tangent point but with a perpendicular tangent line at that point.
783:{\displaystyle \operatorname {isopt} (\theta )=\left\{X\ {\Biggl |}\ \measuredangle {\biggl (}{\overrightarrow {XC}},{\overrightarrow {XD}}{\biggr )}=\theta +2k\pi \right\}.}
915:{\displaystyle {\text{full red circle}}=\left\{X\ {\Biggl |}\ \measuredangle {\biggl (}{\overrightarrow {XC}},{\overrightarrow {XD}}{\biggr )}=\theta +k\pi \right\}}
338:
384:
1435:
540:
1029:
It is relatively easy to show using inversion that, in the
Apollonian circles, every blue circle intersects every red circle orthogonally, i.e., at a
106:
2539:
240:
2139:
51:
Some
Apollonian circles. Every blue circle intersects every red circle at a right angle. Every red circle passes through the two points
1060:
Alternatively, the orthogonal property of the two pencils follows from the defining property of the radical axis, that from any point
2617:
1920:
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1428:
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2194:
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2159:
1057:, it preserves the angles between the curves it transforms, so the original Apollonian circles also meet at right angles.
2527:
1930:
2594:
1421:
2713:
1372:
1301:
806:. When we really want the whole red circle, a description using oriented angles of straight lines has to be used:
947:(red family of circles in the figure) that is defined by two generators that pass through each other in exactly
2563:
2496:
2129:
2009:
17:
1267:
959:(blue family of circles in the figure) that is defined by two generators that do not intersect each other at
930:
1041:. The same inversion transforms the red circles into a set of straight lines that all contain the image of
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2389:
2277:
1452:
509:
2341:
2272:
1289:
1968:
1784:
1759:
650:, a method is required to specify which point is the right one. An isoptic arc is the locus of points
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1834:
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1359:
629:
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as its radical axis. The centers of the circles in this pencil lie on the perpendicular bisector of
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1988:
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1365:, Oxford University Press, esp. Ch. 2 "Harmonic division and Apollonian circles", pp. 13â23
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2232:
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1998:
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A given blue circle and a given red circle intersect in two points. In order to obtain bipolar
354:
Each circle in the first family (the blue circles in the figure) is associated with a positive
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1614:
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Each circle in the second family (the red circles in the figure) is associated with an angle
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37:
1315:
Pfeifer, Richard E.; Van Hook, Cathleen (1993), "Circles, Vectors, and Linear
Algebra",
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75:
such that every circle in the first family intersects every circle in the second family
36:, and the corresponding family of orthogonal circles. For other circles associated with
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1033:. Inversion of the blue Apollonian circles with respect to a circle centered on point
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2016:
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508:. The equation defining these circles as a locus can be generalized to define the
2199:
2189:
2083:
1809:
524:
1053:. Obviously, the transformed pencils meet at right angles. Since inversion is a
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2282:
2174:
2154:
1982:
1539:
1509:
1006:(the blue circles) has its radical axis on the perpendicular bisector of line
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230:{\displaystyle {\frac {|CT_{i}|}{|DT_{i}|}}={\frac {|CT_{o}|}{|DT_{o}|}}=r}
33:
1975:
1863:
1569:
1554:
1037:
results in a pencil of concentric circles centered at the image of point
1030:
470:{\displaystyle \left\{X\ {\Biggl |}\ {\frac {d(X,C)}{d(X,D)}}=r\right\}.}
355:
306:{\displaystyle \angle T_{i}XT_{o}={\frac {180^{\circ }}{2}}=90^{\circ }}
1961:
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Such an arc is contained into a red circle and is bounded by points
1851:
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1226:
1018:
Inversive geometry, orthogonal intersection, and coordinate systems
979:
centers. Any three or more circles from the same family are called
88:
60:
600:{\displaystyle \left\{X\ {\Bigl |}\ C{\hat {X}}D=\theta \right\}.}
504:, the circle degenerates to a line, the perpendicular bisector of
2424:
1749:
1744:
1644:
1634:
1609:
1076:
are all equal. It follows from this that the circle centered at
617:
generates the set of all circles passing through the two points
1719:
1594:
1084:
perpendicularly. The same construction can be applied for each
72:
990:
The elliptic pencil of circles passing through the two points
343:
The
Apollonian circles are defined in two different ways by a
2095:
2073:
1729:
1080:
with length equal to these tangents crosses all circles of
794:. The remaining part of the corresponding red circle is
1385:
1210:"Full-Bloch beams and ultrafast Rabi-rotating vortices"
971:
Any two of these circles within a pencil have the same
939:. Each is determined by any two of its members, called
628:
The two points where all the red circles cross are the
1092:, forming another pencil of circles perpendicular to
994:(the set of red circles, in the figure) has the line
812:
664:
543:
387:
323:
243:
109:
481:
close to zero, the corresponding circle is close to
32:
This article is about a family of circles sharing a
79:, and vice versa. These circles form the basis for
914:
782:
599:
469:
332:
305:
229:
103:Apollonian circle, the angle bisectors in X yield
1125:on the real space where the observation is made.
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55:, and every blue circle separates the two points.
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1314:
935:Both of the families of Apollonian circles are
1026:parabolic pencil is another parabolic pencil.
658:under a given oriented angle of vectors i.e.
1443:
1429:
1284:
1195:
1158:
931:Pencil (mathematics) § Pencil of circles
317:X is located on a half circle with diameter
1436:
1422:
1002:. The hyperbolic pencil defined by points
1262:Akopyan, A. V.; Zaslavsky, A. A. (2007),
1243:
1225:
1049:defined by the Apollonian circles into a
1207:
632:of pairs of circles in the blue family.
519:, and is defined as the locus of points
361:, and is defined as the locus of points
98:
46:
1113:waves. The trajectories arise from the
1045:. Thus, this inversion transforms the
943:of the pencil. Specifically, one is an
493:, the corresponding circle is close to
14:
2696:
2465:Latin translations of the 12th century
1353:
1343:
635:
365:such that the ratio of distances from
2195:Straightedge and compass construction
1417:
1386:
1010:, and all its circle centers on line
975:, and all circles in the pencil have
924:
27:Circles in two perpendicular families
2160:Incircle and excircles of a triangle
1266:, Mathematical World, vol. 26,
512:of larger sets of weighted points.
24:
1308:
244:
25:
2725:
1379:
1068:the lengths of the tangents from
2677:
2664:
1407:Advanced High-School Mathematics
1245:10.1103/PhysRevResearch.3.013007
1064:on the radical axis of a pencil
1170:MathWorld uses âcoaxal,â while
2497:A History of Greek Mathematics
2010:The Quadrature of the Parabola
1208:Dominici; et al. (2021),
1201:
1189:
1184:Akopyan & Zaslavsky (2007)
1177:
1172:Akopyan & Zaslavsky (2007)
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1:
1294:, University of Toronto Press
1268:American Mathematical Society
1255:
1198:, pp. 30â31, Theorem A).
967:Radical axis and central line
497:; for the intermediate value
94:
2278:Intersecting secants theorem
7:
2273:Intersecting chords theorem
2140:Doctrine of proportionality
1291:Geometry of Complex Numbers
1128:
10:
2730:
1969:On the Sphere and Cylinder
1922:On the Sizes and Distances
1348:, Springer, pp. 40â43
1103:
928:
639:
83:. They were discovered by
31:
2671:Ancient Greece portal
2660:
2610:
2488:
2475:Philosophy of mathematics
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2390:Ptolemy's table of chords
2334:
2316:
2215:
2208:
2064:
2026:
1843:
1451:
1445:Ancient Greek mathematics
1047:bipolar coordinate system
510:FermatâApollonius circles
2714:Euclidean plane geometry
2342:Aristarchus's inequality
1915:On Conoids and Spheroids
1214:Physical Review Research
1145:
1123:stereographic projection
1055:conformal transformation
2450:Ancient Greek astronomy
2263:Inscribed angle theorem
2253:Greek geometric algebra
1908:Measurement of a Circle
1367:. Dover reprint, 1990,
1344:Samuel, Pierre (1988),
1296:. Dover reprint, 1979,
1088:on the radical axis of
1051:polar coordinate system
2684:Mathematics portal
2470:Non-Euclidean geometry
2425:Mouseion of Alexandria
2298:Tangent-secant theorem
2248:Geometric mean theorem
2233:Exterior angle theorem
2228:Angle bisector theorem
1932:On Sizes and Distances
1361:Excursions in Geometry
916:
784:
601:
485:, while for values of
471:
340:
334:
307:
231:
56:
2372:Pappus's area theorem
2308:Theorem of the gnomon
2185:Quadratrix of Hippias
2108:Circles of Apollonius
2056:Problem of Apollonius
2034:Constructible numbers
1858:Archimedes Palimpsest
917:
785:
602:
472:
335:
308:
232:
102:
50:
42:circles of Apollonius
2588:prehistoric counting
2385:Ptolemy's inequality
2326:Apollonius's theorem
2165:Method of exhaustion
2135:Diophantine equation
2125:Circumscribed circle
1942:On the Moving Sphere
1317:Mathematics Magazine
810:
662:
541:
385:
321:
241:
107:
2709:Elementary geometry
2674: •
2480:Neusis construction
2400:Spiral of Theodorus
2293:Pythagorean theorem
2238:Euclidean algorithm
2180:Lune of Hippocrates
2049:Squaring the circle
1805:Theon of Alexandria
1480:Aristaeus the Elder
1404:David B. Surowski:
1346:Projective Geometry
1286:Schwerdtfeger, Hans
1236:2021PhRvR...3a3007D
1196:Schwerdtfeger (1962
1159:Schwerdtfeger (1962
1135:Apollonius of Perga
642:Bipolar coordinates
636:Bipolar coordinates
87:, a renowned Greek
85:Apollonius of Perga
81:bipolar coordinates
38:Apollonius of Perga
2367:Menelaus's theorem
2357:Irrational numbers
2170:Parallel postulate
2145:Euclidean geometry
2113:Apollonian circles
1655:Isidore of Miletus
1388:Weisstein, Eric W.
1355:Ogilvy, C. Stanley
1270:, pp. 57â62,
1264:Geometry of Conics
1072:to each circle in
955:). The other is a
937:pencils of circles
925:Pencils of circles
912:
780:
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341:
333:{\displaystyle CD}
330:
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67:are two families (
65:Apollonian circles
57:
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2655:
2408:
2407:
2395:Ptolemy's theorem
2268:Intercept theorem
2118:Apollonian gasket
2044:Doubling the cube
2017:The Sand Reckoner
1277:978-0-8218-4323-9
1174:prefer âcoaxial.â
1161:, pp. 8â10).
1140:Greek mathematics
957:hyperbolic pencil
883:
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654:that sees points
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16:(Redirected from
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2430:Platonic Academy
2377:Problem II.8 of
2347:Crossbar theorem
2303:Thales's theorem
2243:Euclid's theorem
2213:
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2130:Commensurability
2091:Axiomatic system
2039:Angle trisection
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2352:Heron's formula
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2200:Triangle center
2190:Regular polygon
2067:and definitions
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1787:
1782:
1777:
1772:
1767:
1762:
1757:
1752:
1747:
1742:
1737:
1732:
1727:
1722:
1717:
1712:
1707:
1702:
1697:
1692:
1687:
1682:
1677:
1672:
1667:
1662:
1657:
1652:
1647:
1642:
1637:
1632:
1627:
1622:
1617:
1612:
1607:
1602:
1597:
1592:
1587:
1582:
1577:
1572:
1567:
1562:
1557:
1552:
1547:
1542:
1537:
1532:
1527:
1522:
1517:
1512:
1507:
1502:
1497:
1492:
1487:
1482:
1477:
1472:
1467:
1461:
1459:
1453:Mathematicians
1449:
1448:
1441:
1440:
1433:
1426:
1418:
1412:
1411:
1402:
1381:
1380:External links
1378:
1377:
1376:
1351:
1341:
1310:
1307:
1306:
1305:
1282:
1276:
1257:
1254:
1251:
1250:
1200:
1188:
1176:
1163:
1150:
1149:
1147:
1144:
1143:
1142:
1137:
1130:
1127:
1105:
1102:
1019:
1016:
985:coaxal circles
968:
965:
929:Main article:
926:
923:
910:
906:
903:
900:
897:
894:
889:
882:
878:
875:
869:
864:
860:
857:
849:
844:
836:
828:
824:
820:
779:
775:
771:
768:
765:
762:
759:
756:
751:
744:
740:
737:
731:
726:
722:
719:
711:
706:
698:
690:
686:
682:
679:
676:
673:
670:
667:
640:Main article:
637:
634:
596:
592:
588:
585:
582:
576:
573:
567:
559:
551:
547:
523:such that the
477:For values of
466:
462:
458:
455:
449:
446:
443:
440:
437:
434:
429:
426:
423:
420:
417:
414:
403:
395:
391:
329:
326:
300:
296:
292:
287:
282:
278:
272:
267:
263:
259:
254:
250:
246:
226:
223:
216:
210:
206:
202:
198:
191:
185:
181:
177:
173:
166:
159:
153:
149:
145:
141:
134:
128:
124:
120:
116:
96:
93:
26:
18:Coaxal circles
9:
6:
4:
3:
2:
2726:
2715:
2712:
2710:
2707:
2705:
2702:
2701:
2699:
2686:
2685:
2680:
2673:
2672:
2659:
2649:
2646:
2644:
2641:
2639:
2636:
2634:
2631:
2629:
2626:
2624:
2621:
2619:
2616:
2615:
2613:
2609:
2601:
2598:
2597:
2596:
2593:
2589:
2586:
2585:
2584:
2581:
2577:
2574:
2573:
2572:
2569:
2565:
2562:
2561:
2560:
2557:
2553:
2550:
2549:
2548:
2545:
2541:
2538:
2537:
2536:
2533:
2529:
2526:
2525:
2524:
2521:
2517:
2514:
2513:
2512:
2509:
2505:
2501:
2500:
2499:
2498:
2494:
2493:
2491:
2487:
2481:
2478:
2476:
2473:
2471:
2468:
2466:
2463:
2461:
2458:
2456:
2453:
2451:
2448:
2447:
2444:
2441:
2437:
2431:
2428:
2426:
2423:
2421:
2418:
2417:
2415:
2411:
2401:
2398:
2396:
2393:
2391:
2388:
2386:
2383:
2381:
2380:
2375:
2373:
2370:
2368:
2365:
2363:
2360:
2358:
2355:
2353:
2350:
2348:
2345:
2343:
2340:
2339:
2337:
2333:
2327:
2324:
2323:
2321:
2319:
2315:
2309:
2306:
2304:
2301:
2299:
2296:
2294:
2291:
2289:
2288:Pons asinorum
2286:
2284:
2281:
2279:
2276:
2274:
2271:
2269:
2266:
2264:
2261:
2259:
2258:Hinge theorem
2256:
2254:
2251:
2249:
2246:
2244:
2241:
2239:
2236:
2234:
2231:
2229:
2226:
2225:
2223:
2221:
2220:
2214:
2211:
2207:
2201:
2198:
2196:
2193:
2191:
2188:
2186:
2183:
2181:
2178:
2176:
2173:
2171:
2168:
2166:
2163:
2161:
2158:
2156:
2153:
2151:
2148:
2146:
2143:
2141:
2138:
2136:
2133:
2131:
2128:
2126:
2123:
2119:
2116:
2114:
2111:
2110:
2109:
2106:
2104:
2101:
2097:
2094:
2093:
2092:
2089:
2085:
2082:
2080:
2077:
2076:
2075:
2072:
2071:
2069:
2063:
2057:
2054:
2050:
2047:
2045:
2042:
2040:
2037:
2036:
2035:
2032:
2031:
2029:
2025:
2019:
2018:
2014:
2012:
2011:
2007:
2005:
2001:
1997:
1995:
1991:
1987:
1985:
1984:
1980:
1978:
1977:
1973:
1971:
1970:
1966:
1964:
1963:
1959:
1957:
1953:
1949:
1947:
1943:
1939:
1937:
1933:
1929:
1927:
1925:(Aristarchus)
1923:
1919:
1917:
1916:
1912:
1910:
1909:
1905:
1903:
1899:
1895:
1893:
1889:
1885:
1883:
1882:
1878:
1876:
1872:
1868:
1866:
1865:
1861:
1859:
1856:
1854:
1853:
1849:
1848:
1846:
1842:
1836:
1833:
1831:
1830:Zeno of Sidon
1828:
1826:
1823:
1821:
1818:
1816:
1813:
1811:
1808:
1806:
1803:
1801:
1798:
1796:
1793:
1791:
1788:
1786:
1783:
1781:
1778:
1776:
1773:
1771:
1768:
1766:
1763:
1761:
1758:
1756:
1753:
1751:
1748:
1746:
1743:
1741:
1738:
1736:
1733:
1731:
1728:
1726:
1723:
1721:
1718:
1716:
1713:
1711:
1708:
1706:
1703:
1701:
1698:
1696:
1693:
1691:
1688:
1686:
1683:
1681:
1678:
1676:
1673:
1671:
1668:
1666:
1663:
1661:
1658:
1656:
1653:
1651:
1648:
1646:
1643:
1641:
1638:
1636:
1633:
1631:
1628:
1626:
1623:
1621:
1618:
1616:
1613:
1611:
1608:
1606:
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1586:
1583:
1581:
1578:
1576:
1573:
1571:
1568:
1566:
1563:
1561:
1558:
1556:
1553:
1551:
1548:
1546:
1543:
1541:
1538:
1536:
1533:
1531:
1528:
1526:
1523:
1521:
1518:
1516:
1513:
1511:
1508:
1506:
1503:
1501:
1498:
1496:
1493:
1491:
1488:
1486:
1483:
1481:
1478:
1476:
1473:
1471:
1468:
1466:
1463:
1462:
1460:
1458:
1454:
1450:
1446:
1439:
1434:
1432:
1427:
1425:
1420:
1419:
1416:
1409:
1408:
1403:
1399:
1398:
1393:
1389:
1384:
1383:
1374:
1373:0-486-26530-7
1370:
1363:
1362:
1356:
1352:
1347:
1342:
1338:
1334:
1330:
1326:
1322:
1318:
1313:
1312:
1303:
1302:0-486-63830-8
1299:
1293:
1292:
1287:
1283:
1279:
1273:
1269:
1265:
1260:
1259:
1246:
1241:
1237:
1233:
1228:
1223:
1220:(1): 013007,
1219:
1215:
1211:
1204:
1197:
1192:
1185:
1180:
1173:
1167:
1160:
1155:
1151:
1141:
1138:
1136:
1133:
1132:
1126:
1124:
1120:
1116:
1115:Rabi rotation
1112:
1101:
1097:
1058:
1056:
1052:
1048:
1032:
1027:
1024:
1015:
988:
986:
982:
978:
974:
964:
962:
958:
950:
946:
942:
938:
932:
922:
908:
904:
901:
898:
895:
892:
880:
876:
873:
867:
862:
858:
855:
842:
826:
822:
818:
803:
799:
777:
773:
769:
766:
763:
760:
757:
754:
742:
738:
735:
729:
724:
720:
717:
704:
688:
684:
680:
674:
668:
665:
649:
643:
633:
631:
626:
616:
607:
594:
590:
586:
583:
580:
571:
565:
549:
545:
531:
526:
513:
511:
501:
464:
460:
456:
453:
444:
441:
438:
432:
424:
421:
418:
412:
393:
389:
357:
352:
346:
327:
324:
316:
298:
294:
290:
285:
280:
276:
270:
265:
261:
257:
252:
248:
224:
221:
208:
204:
200:
183:
179:
175:
164:
151:
147:
143:
126:
122:
118:
101:
92:
90:
86:
82:
78:
74:
70:
66:
62:
49:
43:
39:
35:
30:
19:
2675:
2662:
2504:Thomas Heath
2495:
2378:
2362:Law of sines
2218:
2150:Golden ratio
2112:
2015:
2008:
1999:
1993:(Theodosius)
1989:
1981:
1974:
1967:
1960:
1951:
1941:
1935:(Hipparchus)
1931:
1921:
1913:
1906:
1897:
1887:
1879:
1874:(Apollonius)
1870:
1862:
1850:
1825:Zeno of Elea
1585:Eratosthenes
1575:Dionysodorus
1406:
1395:
1360:
1345:
1323:(2): 75â86,
1320:
1316:
1290:
1263:
1217:
1213:
1203:
1191:
1179:
1166:
1154:
1119:Bloch sphere
1107:
1098:
1059:
1028:
1021:
989:
984:
980:
973:radical axis
970:
960:
956:
948:
944:
940:
936:
934:
801:
797:
647:
645:
627:
614:
608:
529:
514:
499:
353:
345:line segment
342:
77:orthogonally
64:
58:
34:radical axis
29:
2571:mathematics
2379:Arithmetica
1976:Ostomachion
1945:(Autolycus)
1864:Arithmetica
1640:Hippocrates
1570:Dinostratus
1555:Dicaearchus
1485:Aristarchus
1031:right angle
648:coordinates
356:real number
2698:Categories
2623:Babylonian
2523:arithmetic
2489:History of
2318:Apollonius
2003:(Menelaus)
1962:On Spirals
1881:Catoptrics
1820:Xenocrates
1815:Thymaridas
1800:Theodosius
1785:Theaetetus
1765:Simplicius
1755:Pythagoras
1740:Posidonius
1725:Philonides
1685:Nicomachus
1680:Metrodorus
1670:Menaechmus
1625:Hipparchus
1615:Heliodorus
1565:Diophantus
1550:Democritus
1530:Chrysippus
1500:Archimedes
1495:Apollonius
1465:Anaxagoras
1457:(timeline)
1256:References
1227:1801.02580
941:generators
613:from 0 to
95:Definition
2084:Inscribed
1844:Treatises
1835:Zenodorus
1795:Theodorus
1770:Sosigenes
1715:Philolaus
1700:Oenopides
1695:Nicoteles
1690:Nicomedes
1650:Hypsicles
1545:Ctesibius
1535:Cleomedes
1520:Callippus
1505:Autolycus
1490:Aristotle
1470:Anthemius
1397:MathWorld
1111:polariton
977:collinear
905:π
896:θ
881:→
863:→
843:∡
770:π
758:θ
743:→
725:→
705:∡
675:θ
669:
609:Scanning
587:θ
575:^
489:close to
299:∘
281:∘
245:∠
2648:Japanese
2633:Egyptian
2576:timeline
2564:timeline
2552:timeline
2547:geometry
2540:timeline
2535:calculus
2528:timeline
2516:timeline
2219:Elements
2065:Concepts
2027:Problems
2000:Spherics
1990:Spherics
1955:(Euclid)
1901:(Euclid)
1898:Elements
1891:(Euclid)
1852:Almagest
1760:Serenus
1735:Porphyry
1675:Menelaus
1630:Hippasus
1605:Eutocius
1580:Domninus
1475:Archytas
1357:(1969),
1288:(1962),
1186:, p. 59.
1129:See also
1121:and its
951:points (
347:denoted
89:geometer
61:geometry
2704:Circles
2628:Chinese
2583:numbers
2511:algebra
2439:Related
2413:Centers
2209:Results
2079:Central
1750:Ptolemy
1745:Proclus
1710:Perseus
1665:Marinus
1645:Hypatia
1635:Hippias
1610:Geminus
1600:Eudoxus
1590:Eudemus
1560:Diocles
1410:. p. 31
1337:2691113
1232:Bibcode
1117:of the
1104:Physics
963:point.
533:equals
377:equals
373:and to
73:circles
69:pencils
2643:Indian
2420:Cyrene
1952:Optics
1871:Conics
1790:Theano
1780:Thales
1775:Sporus
1720:Philon
1705:Pappus
1595:Euclid
1525:Carpus
1515:Bryson
1371:
1335:
1300:
1274:
840:
830:
802:π
798:θ
796:isopt(
702:
692:
563:
553:
407:
397:
237:, due
40:, see
2638:Incan
2559:logic
2335:Other
2103:Chord
2096:Axiom
2074:Angle
1730:Plato
1620:Heron
1540:Conon
1333:JSTOR
1222:arXiv
1146:Notes
666:isopt
71:) of
2600:list
1888:Data
1660:Leon
1510:Bion
1369:ISBN
1298:ISBN
1272:ISBN
1004:C, D
992:C, D
953:C, D
792:C, D
656:C, D
621:and
313:and
53:C, D
2502:by
2216:In
1325:doi
1240:doi
983:or
961:any
949:two
530:CXD
502:= 1
369:to
277:180
59:In
2700::
1394:,
1390:,
1331:,
1321:66
1319:,
1238:,
1230:,
1216:,
1212:,
1096:.
1014:.
1012:CD
1008:CD
1000:CD
996:CD
987:.
800:+
625:.
537:,
506:CD
381:,
351:.
349:CD
295:90
91:.
63:,
1437:e
1430:t
1423:v
1375:.
1350:.
1340:.
1327::
1304:.
1281:.
1242::
1234::
1224::
1218:3
1094:P
1090:P
1086:X
1082:P
1078:X
1074:P
1070:X
1066:P
1062:X
1043:D
1039:D
1035:C
909:}
902:k
899:+
893:=
888:)
877:D
874:X
868:,
859:C
856:X
848:(
835:|
827:X
823:{
819:=
804:)
778:.
774:}
767:k
764:2
761:+
755:=
750:)
739:D
736:X
730:,
721:C
718:X
710:(
697:|
689:X
685:{
681:=
678:)
672:(
652:X
623:D
619:C
615:Ď
611:θ
595:.
591:}
584:=
581:D
572:X
566:C
558:|
550:X
546:{
535:θ
528:â
521:X
517:θ
500:r
495:D
491:â
487:r
483:C
479:r
465:.
461:}
457:r
454:=
448:)
445:D
442:,
439:X
436:(
433:d
428:)
425:C
422:,
419:X
416:(
413:d
402:|
394:X
390:{
379:r
375:D
371:C
367:X
363:X
359:r
328:D
325:C
291:=
286:2
271:=
266:o
262:T
258:X
253:i
249:T
225:r
222:=
215:|
209:o
205:T
201:D
197:|
190:|
184:o
180:T
176:C
172:|
165:=
158:|
152:i
148:T
144:D
140:|
133:|
127:i
123:T
119:C
115:|
44:.
20:)
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