X3720

X756com(INCENTRAL TRIANGLE)

X3720 = bicentric sum of PU(84)
< 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 { 37, 38 }: Central Line L(37, 38) { 81, 238 }: Central Line L(81, 238) { 86, 310 }: Central Line L(86, 310) { 6, 748 }: Central Line L(6, 748) { 55, 750 }: Central Line L(55, 750) 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_2 = CENTROID (0.5374, 0.2000) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5567, 0.2158) X_38 = CROSSPOINT OF X(1) AND X(75) (0.5443, 0.1976) X_81 = CEVAPOINT OF INCENTER AND SYMMEDIAN POINT (0.5849, 0.2510) X_238 = -LINE CONJUGATE OF X(37) (1.6190, 1.4775) X_86 = CEVAPOINT OF INCENTER AND CENTROID (0.5534, 0.2171) X_310 = ISOTOMIC CONJUGATE OF X(42) (0.4762, 0.1536) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_748 = INTERSECTION OF LINES X(2)X(31) AND X(9)X(38) (0.4360, 0.0802) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_750 = INTERSECTION OF LINES X(1)X(88) AND X(2)X(31) (0.5675, 0.2354) X_3720 = com(INCENTRAL TRIANGLE) (0.5630, 0.2251) A B C X1 X2 X37 X38 X81 X238 X86 X310 X6 X748 X55 X750 X3720 (a, b, c) = (5, 6, 7) | AB = 1

X3720 (X756com(INCENTRAL TRIANGLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Barycentrics: \(a(ab + ac + 2bc) : :\)
Search Value (6, 9, 13): \(1.908160564542\)
▸ Incident Central Lines34 lines

{1, 2}, {6, 748}, {11, 3136}, {31, 940}, {37, 38}, {55, 750}, {56, 1011}, {57, 968}, {73, 1985}, {77, 2898}, {81, 238}, {86, 310}, {171, 902}, {226, 1458}, {244, 1962}, {291, 1255}, {405, 1468}, {427, 2356}, {497, 2293}, {518, 756}, {649, 2666}, {799, 2107}, {846, 3218}, {851, 2646}, {942, 2292}, {960, 2650}, {991, 1699}, {1042, 3485}, {1100, 2238}, {1197, 3231}, {1279, 2239}, {1376, 2177}, {1818, 2886}, {1836, 3000}

▸ Functional Relations4 relations

Notes

Let IaIbIc be the incentral triangle. Let A″ be the intersection, other than the midpoint of BC, of the A-median and the bicevian ellipse of X1 and X2. Define B″ and C″ cyclically. The lines IaA″, IbB″, IcC″ concur in X3720. (Randy Hutson, December 26, 2015)