X1601–X1661

TCC-Perspectors & Tripolar Centroids

Perspectors of the tangential triangle and the circumcevian triangle of P (X1601–X1634), tripolar centroids (X1635–X1651), and Evans and Grinberg perspectors.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) ( A R_b ): Euclid I.22 Guide Circle (A) ( B R_a ): Euclid I.22 Guide Circle (B) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) circumcircle: Circumcircle of △ABC incircle: Incircle of △ABC A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) R_b = Radius Point R_b (0.8571, 0.0000) R_a = Radius Point R_a (0.2857, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1 = INCENTER (0.5714, 0.2333) X_1601 = TCC-PERSPECTOR OF X(5) (0.4370, 0.2016) X_1602 = TCC-PERSPECTOR OF X(7) (0.5590, 0.3060) X_1604 = TCC-PERSPECTOR OF X(9) (0.3886, -0.5005) X_1606 = TCC-PERSPECTOR OF X(14) (0.8673, -0.5199) X_1609 = TCC-PERSPECTOR OF X(19) (0.7601, 0.5316) X_1610 = TCC-PERSPECTOR OF X(21) (0.4119, -0.3400) X_1613 = TCC-PERSPECTOR OF X(31) (0.7015, 0.3763) X_1614 = TCC-PERSPECTOR OF X(54) (0.5582, -0.0024) X_1615 = TCC-PERSPECTOR OF X(55) (0.4519, -0.2046) X_1618 = TCC-PERSPECTOR OF X(59) (0.0333, 0.7076) X_1621 = TCC-PERSPECTOR OF X(81) (0.5431, 0.1961) X_1623 = TCC-PERSPECTOR OF X(88) (0.9310, -0.3379) X_1624 = TCC-PERSPECTOR OF X(162) (0.4025, 0.6319) X_1628 = TCC-PERSPECTOR OF X(254) (0.4502, -0.0283) X_1629 = TCC-PERSPECTOR OF X(275) (0.6231, 0.4755) X_1632 = TCC-PERSPECTOR OF X(648) (0.2937, 0.6477) A B C X3 X1 X1601 X1602 X1604 X1606 X1609 X1610 X1613 X1614 X1615 X1618 X1621 X1623 X1624 X1628 X1629 X1632 (a, b, c) = (5, 6, 7) | AB = 1

X1601–X1661 (TCC-Perspectors & Tripolar Centroids) — 16 representative centers (from 61 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — TCC Perspectors 1601-1634

Suppose P is a point. As noted in TCCT, p. 201, the tangential triangle is perspective to the circumcevian triangle of P. (The tangential triangle is also perspector to the circum-anticevian triangle of P, with the same perspector as for the circumcevian triangle of P.) In August 2003, Jean-Pierre Ehrmann noted that if P =\(x : y : z\) (barycentrics), then the perspector is given by

\(a^{2}(b^{4}/y^{2} + c^{4}/z^{2} - a^{4}/x^{2})\) : \(b^{2}(c^{4}/z^{2} + a^{4}/x^{2} - b^{4}/y^{2})\) : \(c^{2}(a^{4}/x^{2} + b^{4}/y^{2} - c^{4}/z^{2})\) .

Denote this perspector by T(P), and call it the TCC-perspector of P. If P =\(x : y : z\) (trilinears), then

T(P) =\(a(b^{2}/y^{2} + c^{2}/z^{2} - a^{2}/x^{2}) : b(c^{2}/z^{2} + a^{2}/x^{2} - b^{2}/y^{2}) : c(a^{2}/x^{2} + b^{2}/y^{2} - c^{2}/z^{2})\).

The transformation T carries triangle centers to triangle centers. The appearance of i → j in the following list means that T(Xi) = Xj: 1 → 3, 2 → 22, 3 → 1498, 4 → 24, 5 → 1601, 6 → 6, 7 → 1602, 8 → 1603, 9 → 1604, 13 → 1605, 14 → 1606, 15 → 24303, 17 → 1607, 18 → 1608, 19 → 1609, 21 → 1610, 25 → 1611, 28 → 1612, 31 → 1613, 32 → 33786, 35 → 33669, 54 → 1614, 55 → 1615, 56 → 1616, 57 → 1617, 58 → 595, 59 → 1618, 63 → 1619, 64 → 1620, 75 → 33801, 76 → 33802, 81 → 1621, 83 → 1078, 84 → 1622, 86 → 23374, 88 → 1623, 162 → 1624, 163 → 1625, 174 → 1626, 188 → 2933, 249 → 33803, 251 → 1627, 254 → 1628, 259 → 198, 266, → 56, 275 → 1629, 284 → 1630, 365 → 55, 366 → 1631, 508 → 23852, 509 → 1486, 512 → 33704, 523 → 30715, 648 → 1632, 651 → 1633, 662 → 1634, 1126 → 33771, 1171 → 33774, 2153 → 11142, 2154 → 11141, 3445 → 33804, 5374 → 159, 6727 → 23846, 6733 → 23845, 14085 → 101, 14086 → 30715, 14089 → 99, 14090 → 33704, 18297 → 23849, 18753 → 2176, 20034 → 25

Group Preamble — Tripolar Centroids 1635-1651

Suppose P, Q, R are collinear points not on the line at infinity. Let M = midpoint{Q, R}. The segment PM has two trisectors. The trisector closer to M here defines the centroid of {P, Q, R}.

Suppose X =\(x : y : z\) (trilinears) is a point other than X2, and define the tripolar centroid of X as the point TG(X) given by

TG(X) = x(by - cz)(by + cz - 2ax) : y(cz - ax)(cz + ax - 2by) : z(ax - by)(ax + by - 2cz).

TG(X) is the centroid of the points BC∩B'C', CA∩C'A', AB∩A'B', where A'B'C' denotes the cevian triangle of X. The notions of centroid and tripolar centroid were contributed by Darij Grinberg, August 24, 2003.

The appearance of (i,j) in the following list means that TG(Xi) = Xj:

(1,1635), (3,1636), (4,1637), (5,14391), (6,351), (7,1638), (8,1639), (9,14392), (10,4120), (11,14393), (12,14394), (13,9200), (14,9201), (17,14446), (18,14447), (21,14395), (22,14396), (24,14397), (25,14398), (27,11125), (28,14399), (29,14400), (30,14401), (31,14402), (32,14403), (37,14404), (38,14405), (39,14406), (42,14407), (43,14408), (44,14409), (45,14410), (55,14411), (56,14412), (57,14413), (63,14414), (65,14415), (66,14417), (69,14417), (75,4728), (76,9148), (78,14418), (81,14419), (83,14420), (86,4750), (88,14421), (89,14422), (98,1640), (99,1641}, {100,1642), (105,1643), (111,9171), (115,14423), (141,14424), (145,14425), (190,1644), (192,14426), (200,14427), (239,4448), (251,14428), (262,3569), (263,2491), (298,9204), (299,9205), (306,14429), (312,14430), (321,14431), (333,14432), (350,14433), (385,11183), (512,1645), (513,1646), (514,1647}, {519,6544), (523,1648), (524,1649), (525,1650), (536,14434), (551,14435), (648,1651), (671,8371), (869,14436), (899,14437), (957,3310), (985,14438), (1002,665), (1022,244), (1125,4984), (1026,14439), (1267,14440), (1646,14441), (1647,14442), (1648,14443), (1649,14444), (1698,4958), (1916,11182), (1976,6041}, {1992,9125), (2394,125), (2395,6784), (2396,6786), (2403,3756), (2407,5642), (2408,6791), (2409,6793), (2418,12036), (2419,12037), (2421,9155), (3413,13636}, {3414,13722), (3616,4773), (4049,3120), (4240,3163), (5391,14445), (5466,115), (5468,2482), (6548,1086), (9178,3124), (9213,2088), (9221,2081), (13582,1116), (14223,868)

X_1614 — TCC-PERSPECTOR OF X_54

X1614TCC-PERSPECTOR OF X54

> X1614 lies on these lines: 3,74 4,184 6,1173 20,1147 22,155 23,52 24,154 30,49 51,1199 70,1176 185,186 376,1092 378,1498 389,1495 546,567 1503,1594
X_1654 — 1st HATZIPOLAKIS PARALLELIAN POINT

X16541st HATZIPOLAKIS PARALLELIAN POINT

Let P be a point in the plane of, but not on a sideline of, triangle ABC. Let BA be the point where the line through P parallel to line BC meets line BA, and let CA be the point where the line through P parallel to line BC meets line CA. Define CB, AB, AC, and BC cyclically. If P = X1654, then
X_1655 — 2nd HATZIPOLAKIS PARALLELIAN POINT

X16552nd HATZIPOLAKIS PARALLELIAN POINT

Continuing from the description of X1654, let h(B,A) be the distance from the point BA to the line CA, and define five other distances cyclically. If P = X1655, then