X1560

ORTHOJOIN OF X468

X1560 = X2-Ceva conjugate of X468
< 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) { 6, 67 }: Central Line L(6, 67) { 2, 112 }: Central Line L(2, 112) { 4, 111 }: Central Line L(4, 111) { 53, 136 }: Central Line L(53, 136) { 122, 216 }: Central Line L(122, 216) { 115, 427 }: Central Line L(115, 427) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_468 = -LINE CONJUGATE OF X(3) (0.7498, 0.7558) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_67 = ISOGONAL CONJUGATE OF X(23) (0.7606, -0.2152) X_112 = Ψ(ORTHOCENTER, SYMMEDIAN POINT) (0.5162, 0.6121) X_4 = ORTHOCENTER (0.6122, 0.3958) X_111 = PARRY POINT (0.9885, -0.0457) X_53 = SYMMEDIAN POINT OF ORTHIC TRIANGLE (0.6164, 0.4390) X_136 = INTERSECTION OF LINE X(4)X(110) AND X(25)X(132) (0.8024, 0.3155) X_122 = -OF-MEDIAL-TRIANGLE (0.5195, -0.0036) X_216 = -CEVA CONJUGATE OF X(51) (0.5201, 0.1353) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7507, 0.0838) X_427 = COMPLEMENT OF X(22) (0.5990, 0.3610) X_1560 = ORTHOJOIN OF X(468) (0.5219, 0.5018) A B C X2 X468 X6 X67 X112 X4 X111 X53 X136 X122 X216 X115 X427 X1560 (a, b, c) = (5, 6, 7) | AB = 1

X1560 (ORTHOJOIN OF X468) 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, 2373}

  • Related Centers: X2 (CENTROID), X2373 (CIR(X67))

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) (see note above X1512)
Search Value (6, 9, 13): \(-2.983394682623\)

X1560 lies on the nine-point circle.

▸ Incident Central Lines8 lines

{2, 112}, {4, 111}, {6, 67}, {53, 136}, {115, 427}, {122, 216}, {135, 571}, {187, 468}

X1560 lies on hyperbola {{X2,X6,X216,X233,X1249,X1560,X3162}}. This hyperbola is a circumconic of the medial triangle, and the locus of perspectors of circumconics centered at a point on the Euler line. The hyperbola is tangent to Euler line at X2. (Randy Hutson, June 7, 2019)

▸ Functional Relations8 relations