X3817

X2com(3rd EULER TRIANGLE)

X3817 = complement of X165
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 165 }: Central Line L(2, 165) { 5, 10 }: Central Line L(5, 10) { 11, 118 }: Central Line L(11, 118) { 381, 515 }: Central Line L(381, 515) { 262, 726 }: Central Line L(262, 726) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) M_a = Midpoint M_a of BC (0.8061, 0.2999) M_b = Midpoint M_b of CA (0.3061, 0.2999) M_c = Midpoint M_c of AB (0.5000, 0.0000) X_2 = CENTROID (0.5374, 0.2000) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4762, 0.0583) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_118 = MIDPOINT OF X(4) AND X(101) (0.4133, 0.4603) X_381 = MIDPOINT OF X(2) AND X(4) (0.5748, 0.2979) X_515 = ISOGONAL CONJUGATE OF X(102) (0.6804, 0.7691) X_262 = ISOGONAL CONJUGATE OF X(182) (0.5893, 0.2856) X_726 = EVEN (- 2, -1) INFINITY POINT (1.0187, 0.5358) X_3817 = com(3rd EULER TRIANGLE) (0.5680, 0.2708) A B C Ma Mb Mc X2 X165 X5 X10 X11 X118 X381 X515 X262 X726 X3817 (a, b, c) = (5, 6, 7) | AB = 1

X3817 (X2com(3rd EULER TRIANGLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Barycentrics: \(3b^{3} + 3c^{3} - a^{2}b - a^{2}c - 2ab^{2} - 2ac^{2} - 3b^{2}c - 3bc^{2} + 4abc : :\)
Search Value (6, 9, 13): \(-0.295803989155\)
▸ Incident Central Lines25 lines

{1, 3091}, {2, 165}, {4, 1125}, {5, 10}, {11, 118}, {20, 3624}, {40, 3090}, {57, 1776}, {142, 1538}, {262, 726}, {355, 3244}, {381, 515}, {519, 3545}, {546, 1385}, {748, 1754}, {908, 3681}, {944, 3636}, {962, 1698}, {971, 3742}, {1210, 3671}, {1323, 2898}, {1482, 3625}, {1709, 3306}, {3057, 3614}, {3579, 3628}

▸ Functional Relations3 relations

Notes

In the plane of a triangle ABC, let I = X1 and

L = line through midpoint of CA perpendicular to BI

L' = line through midpoint of AB perpendicular to CI

L″ = line through midpoint of AI perpendicular to BC

The lines L, L', L″ concur in a point, A'; define B' and C' cyclically. Ten X3817 = X2-of-A'B'C'. Also, A'B'C' = complement of the excentral triangle, and A'B'C' = extraversion triangle of X10. (Randy Hutson, September 14, 2016)

Let TaTbTc be the intouch triangle, and AaAbAc, BaBbBc, CaCbCc the A-, B-, and C-extouch triangles. Let Ab' = TbTc∩BcBa and Ac' = TbTc∩CaCb. Define Bc' and Ba', Ca', Cb' cyclically. Then X3817 is the centroid of {Ab',Ac',Bc',Ba',Ca',Cb'} is X3817. The six points lie on a common ellipse. (Randy Hutson, September 14, 2016)

Let Wa be the inverter of the B- and C-excircles, and define Wb and Wc cyclically; see X5577. Let Ia be Wa-inverse of the incircle, and define Ib and Ic cyclically. The radical center of circles Ia, Ib, Ic is X3817. (Randy Hutson, September 14, 2016)

Let A'B'C' be the circumcevian triangle of X1. Let L1 be the Simson line of A', and define L2 and L3 cyclically. Let A″ = L2∩L3, and define B″ and C″ cyclically. Then X3817 = X2-of-A″B″C″. (Angel Montesdeoca, June 28, 1017)

X3817 is the centroid of AbAcBcBaCaCb in the construction of the perspeconic of the Ursa-minor and Ursa-major triangles. (Randy Hutson, June 27, 2018)