X3746

X191com[n(MEDIAL)*n(INCENTRAL)]

X3746 = incentral-isogonal conjugate of X1
< 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, 3 }: OI Line (Incenter–Circumcenter Line) { 21, 519 }: Central Line L(21, 519) { 12, 546 }: Central Line L(12, 546) { 79, 516 }: Central Line L(79, 516) { 42, 595 }: Central Line L(42, 595) { 191, 518 }: Central Line L(191, 518) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1 = INCENTER (0.5714, 0.2333) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_21 = SCHIFFLER POINT (0.5287, 0.1771) X_519 = ISOGONAL CONJUGATE OF X(106) (0.9566, 0.6107) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_546 = MIDPOINT OF X(4) AND X(5) (0.5842, 0.3223) X_79 = ISOGONAL CONJUGATE OF X(35) (0.5976, 0.3019) X_516 = ISOGONAL CONJUGATE OF X(103) (0.7703, 0.7386) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.5880, 0.2495) X_595 = 2nd HATZIPOLAKIS-YIU POINT (0.6456, 0.3307) X_191 = -CEVA CONJUGATE OF X(1) (0.4859, 0.1209) X_518 = ISOGONAL CONJUGATE OF X(105) (0.9154, 0.6489) X_3746 = com[n(MEDIAL)*n(INCENTRAL)] (0.5547, 0.2026) A B C X1 X3 X21 X519 X12 X546 X79 X516 X42 X595 X191 X518 X3746 (a, b, c) = (5, 6, 7) | AB = 1

X3746 (X191com[n(MEDIAL)*n(INCENTRAL)]) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(3 + 2 \cos A : 3 + 2 \cos B : 3 + 2 \cos C\) (Peter Moses, 12/19/11)
Barycentrics: \(a^{2}(a^{2} - b^{2} - c^{2} - 3bc)\)
Search Value (6, 9, 13): \(2.361972731682\)
▸ Incident Central Lines40 lines

{1, 3}, {5, 3058}, {10, 1621}, {11, 3628}, {12, 546}, {21, 519}, {42, 595}, {58, 902}, {62, 1250}, {79, 516}, {80, 943}, {100, 1125}, {140, 3582}, {145, 993}, {191, 518}, {238, 3293}, {386, 2177}, {388, 3529}, {390, 1479}, {404, 551}, {442, 528}, {495, 3585}, {496, 632}, {497, 498}, {499, 1058}, {575, 2330}, {576, 3056}, {612, 1995}, {1001, 1698}, {1124, 3594}, {1126, 2308}, {1253, 1497}, {1283, 2292}, {1335, 3592}, {1376, 3624}, {1478, 3146}, {1500, 1914}, {2066, 3301}, {2241, 2276}, {2975, 3244}

▸ Functional Relations3 relations

Notes

Let IaIbIc be the incentral triangle. Let La be the reflection of line BC in line IbIc, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, B″ = Lc∩La, C″ = La∩Lb. The lines IaA″, IbB″, IcC″ concur in X3746. (Randy Hutson, September 14, 2016)