X3647

KS(MEDIAL TRIANGLE)

X3647 = complement of X79
< 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, 79 }: Central Line L(2, 79) { 1, 21 }: Central Line L(1, 21) { 10, 30 }: Central Line L(10, 30) { 124, 128 }: Central Line L(124, 128) { 214, 960 }: Central Line L(214, 960) 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_79 = ISOGONAL CONJUGATE OF X(35) (0.5976, 0.3019) X_1 = INCENTER (0.5714, 0.2333) X_21 = SCHIFFLER POINT (0.5287, 0.1771) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_124 = -OF-MEDIAL-TRIANGLE (0.6243, 0.0030) X_128 = -OF-ORTHIC-TRIANGLE (0.3310, 0.3690) X_214 = -CEVA CONJUGATE OF X(44) (0.3571, 0.3499) X_960 = INTERSECTION OF LINES X(1)X(6) AND X(5)X(10) (0.5041, 0.1533) X_3647 = KS(MEDIAL TRIANGLE) (0.5073, 0.1490) A B C Ma Mb Mc X2 X79 X1 X21 X10 X30 X124 X128 X214 X960 X3647 (a, b, c) = (5, 6, 7) | AB = 1

X3647 (KS(MEDIAL TRIANGLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = (2a + b + c)(b^{2} + c^{2} - a^{2} + bc)\)
Barycentrics: \(h(A,B,C) : h(B,C,A) : h(C,A,B)\), where \(h(A,B,C) = (2 \sin A + \sin B + \sin C)(1 + 2 \cos A)\)
Search Value (6, 9, 13): \(3.920596035744\)
▸ Alternate Coordinates1 form

  • Trilinears: \(g(A,B,C) : g(B, C, A) : g(C, A, B)\), where (csc A) (2 sin A + sin B + sin C)(1 + 2 cos A)

▸ Linear Combinations1 combination
▸ Incident Central Lines13 lines

{1, 21}, {2, 79}, {9, 2173}, {10, 30}, {35, 3219}, {44, 2092}, {45, 2305}, {100, 3065}, {124, 128}, {214, 960}, {442, 1155}, {540, 3178}, {553, 1125}

▸ Functional Relations1 relation

Notes

Let (OA) be the reflection in A of the A-excircle, and define (OB) and (OC) cyclically. X3647 is the radical center of (OA), (OB), (OC). (Randy Hutson, June 7, 2019)