X126

X111-OF-MEDIAL-TRIANGLE

X126 = X111-of-medial triangle.
< 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, 111 }: Central Line L(2, 111) { 2, 99 }: Central Line L(2, 99) { 125, 141 }: Central Line L(125, 141) { 625, 858 }: Central Line L(625, 858) 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_111 = PARRY POINT (0.9885, -0.0457) X_99 = STEINER POINT (0.1109, 0.4322) X_125 = CENTER OF JERABEK HYPERBOLA (0.6803, 0.0260) X_141 = COMPLEMENT OF SYMMEDIAN POINT (0.5061, 0.1663) X_625 = MIDPOINT OF X(623) AND X(624) (0.2068, -0.3280) X_858 = COMPLEMENT OF X(23) (0.1126, -0.9116) X_126 = -OF-MEDIAL-TRIANGLE (0.3119, 0.3228) A B C Ma Mb Mc X2 X111 X99 X125 X141 X625 X858 X126 (a, b, c) = (5, 6, 7) | AB = 1

X126 (X111-OF-MEDIAL-TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

  • Derived Triangle: Medial Triangle △MaMbMc — vertices Ma, Mb, Mc (Formed by the midpoints of sides BC, CA, AB)

  • Central Lines: {2, 111}

  • Related Centers: X2 (CENTROID), X111 (PARRY POINT)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc(2a^{2} - b^{2} - c^{2})[b^{4} + c^{4} + a^{2}(b^{2} + c^{2}) - 4b^{2}c^{2}]\)
Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = (2a^{2} - b^{2} - c^{2})[b^{4} + c^{4} + a^{2}(b^{2} + c^{2}) - 4b^{2}c^{2}]\)
Search Value (6, 9, 13): \(3.340332250420\)

X126 lies on the nine-point circle

▸ Incident Central Lines3 lines

{2, 99}, {125, 141}, {625, 858}

▸ Functional Relations11 relations