X1638

TRIPOLAR CENTROID OF X7

X1638 is the tripolar centroid of X7.
< 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) { 1639, 2 }: Central Line L(1639, 2) { 11, 244 }: Central Line L(11, 244) { 57, 654 }: Central Line L(57, 654) { 241, 514 }: Central Line L(241, 514) { 88, 673 }: Central Line L(88, 673) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1639 = TRIPOLAR CENTROID OF X(8) (-0.6122, -0.5999) X_2 = CENTROID (0.5374, 0.2000) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_244 = -LINE CONJUGATE OF X(100) (0.7857, 0.1166) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_654 = CROSSDIFFERENCE OF X(1) AND X(5) (2.8235, 1.7291) X_241 = -LINE CONJUGATE OF X(55) (1.1217, 1.2442) X_514 = ISOGONAL CONJUGATE OF X(101) (1.0761, -0.0329) X_88 = ISOGONAL CONJUGATE OF X(44) (1.0000, 0.0000) X_673 = TRILINEAR POLE OF LINE X(1)X(514) (0.9471, -0.0769) X_1638 = TRIPOLAR CENTROID OF X(7) (1.6871, 0.9998) A B C X1639 X2 X11 X244 X57 X654 X241 X514 X88 X673 X1638 (a, b, c) = (5, 6, 7) | AB = 1

X1638 (TRIPOLAR CENTROID OF X7) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) (see note above X1635)
Barycentrics: \((b - c) (2 a^{2} - b^{2} - c^{2} - a b - a c + 2 b c) : :\)
Search Value (6, 9, 13): \(-4.757905434027\)
▸ Incident Central Lines7 lines

{2, 918}, {11, 244}, {57, 654}, {88, 673}, {241, 514}, {354, 926}, {651, 658}

▸ Functional Relations4 relations

Notes

Let Oa be the circle centered at A with radius k(b + c -a) (for some constant k>0), and define Ob and Oc cyclically. Let A' be the exsimilicenter of Ob and Oc, and define B' and C' cyclically. The centroid of (degenerate) triangle A'B'C' = X1638. This is independent of the choice of k. (Randy Hutson, January 29, 2018)