X3616

H(X1; M, 1/2)

Let OA be the circle tangent to side BC at its midpoint and to the circumcircle on the side of BC opposite A. Define OB and OC cyclically. Let LA be the external tangent to circles OB and OC that is the nearer of the two to OA. Define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. Then A'B'C' is homothetic to ABC, and the center of homothety is X3616. See the reference at X1001.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 1, 2 }: Nagel Line { 7, 21 }: Central Line L(7, 21) { 29, 278 }: Central Line L(29, 278) { 149, 214 }: Central Line L(149, 214) { 55, 404 }: Central Line L(55, 404) { 142, 390 }: Central Line L(142, 390) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2334 = -ISOCONJUGATE OF X(1449) (0.5898, 0.2535) X_1 = INCENTER (0.5714, 0.2333) X_2 = CENTROID (0.5374, 0.2000) X_7 = GERGONNE POINT (0.5903, 0.2769) X_21 = SCHIFFLER POINT (0.5287, 0.1771) X_29 = CEVAPOINT OF INCENTER AND ORTHOCENTER (0.5903, 0.3384) X_278 = ISOGONAL CONJUGATE OF X(219) (0.6278, 0.4577) X_149 = REFLECTION OF X(20) IN X(104) (1.3265, -0.3333) X_214 = -CEVA CONJUGATE OF X(44) (0.3571, 0.3499) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_404 = {X(2),X(3)}-HARMONIC CONJUGATE OF X(21) (0.5538, 0.2429) X_142 = COMPLEMENT OF X(9) (0.5506, 0.2192) X_390 = REFLECTION OF GERGONNE POINT IN INCENTER (0.5526, 0.1897) X_3616 = H(X(1); M, 1/2) (0.5510, 0.2133) A B C X2334 X1 X2 X7 X21 X29 X278 X149 X214 X55 X404 X142 X390 X3616 (a, b, c) = (5, 6, 7) | AB = 1

X3616 (H(X1; M, 1/2)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(bc(3a + b + c) : :\)
Barycentrics: \(3a + b + c : a + 3b + c : a + b + 3c\)
Search Value (6, 9, 13): \(2.253744679276\)
▸ Alternate Coordinates1 form

  • Trilinears: \(r + R \sin B \sin C : :\)

▸ Linear Combinations1 combination
▸ Incident Central Lines62 lines

{1, 2}, {3, 962}, {4, 1385}, {5, 944}, {7, 21}, {9, 1475}, {11, 2476}, {12, 1388}, {20, 946}, {29, 278}, {37, 2272}, {40, 3306}, {55, 404}, {63, 3333}, {69, 1386}, {85, 3160}, {100, 474}, {104, 3560}, {105, 1036}, {106, 835}, {140, 1482}, {142, 390}, {149, 214}, {226, 452}, {238, 1468}, {244, 986}, {320, 3246}, {329, 405}, {330, 1655}, {346, 3247}, {354, 960}, {355, 3090}, {377, 497}, {388, 1319}, {391, 1449}, {392, 942}, {406, 1870}, {442, 496}, {443, 1058}, {459, 1829}, {515, 3091}, {516, 3522}, {517, 631}, {940, 1191}, {941, 2277}, {952, 1656}, {958, 3304}, {966, 1100}, {968, 988}, {982, 2292}, {1000, 1392}, {1056, 3436}, {1220, 2899}, {1320, 3035}, {1376, 3303}, {1479, 2475}, {1699, 3146}, {1788, 2099}, {1962, 3210}, {3218, 3338}, {3242, 3589}, {3524, 3579}

▸ Functional Relations5 relations

Notes

See Dasari Naga Vijay Krishna, On a Conic Through Twelve Notable Points, Int. J. Adv. Math. and Mech. 7(2) (2019) 1-15.