X121

X106-OF-MEDIAL-TRIANGLE

X121 = X106-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) [ M_a X_121 ]: Line [ M_a X_121 ] [ M_c X_121 ]: Line [ M_c X_121 ] { 10, 11 }: Central Line L(10, 11) { 2, 106 }: Central Line L(2, 106) { 116, 141 }: Central Line L(116, 141) 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_10 = SPIEKER CENTER (0.5204, 0.1833) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_2 = CENTROID (0.5374, 0.2000) X_106 = Λ(INCENTER, CENTROID) (1.0000, 0.0000) X_116 = MIDPOINT OF X(4) AND X(103) (0.6990, 0.0375) X_141 = COMPLEMENT OF SYMMEDIAN POINT (0.5061, 0.1663) X_121 = -OF-MEDIAL-TRIANGLE (0.3061, 0.2999) A B C Ma Mb Mc X10 X11 X2 X106 X116 X141 X121 (a, b, c) = (5, 6, 7) | AB = 1

X121 (X106-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)

  • Concurrence Lines: Ma–X121, Mc–X121 concurring at X121

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc(b + c - 2a)[b^{3} + c^{3} + a(b^{2} + c^{2}) - 2bc(b + c)]\)
Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = (b + c - 2a)[b^{3} + c^{3} + a(b^{2} + c^{2}) - 2bc(b + c)]\)
Search Value (6, 9, 13): \(3.738978994571\)

X121 lies on the nine-point circle

▸ Incident Central Lines3 lines

{2, 106}, {10, 11}, {116, 141}

▸ Functional Relations8 relations