X3627

H⁻¹(X3; M,-1/2)

X3627 = X1482-of-orthic-triangle if ABC is acute
< 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) { 2, 3 }: Euler Line { 143, 185 }: Central Line L(143, 185) { 156, 1514 }: Central Line L(156, 1514) { 576, 1353 }: Central Line L(576, 1353) { 515, 1483 }: Central Line L(515, 1483) { 53, 3284 }: Central Line L(53, 3284) 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_3 = CIRCUMCENTER (0.5000, 0.1021) X_4 = ORTHOCENTER (0.6122, 0.3958) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6358, 0.3534) X_185 = NAGEL POINT OF THE ORTHIC TRIANGLE (0.6033, 0.1643) X_156 = -OF-TANGENTIAL-TRIANGLE (0.6520, -0.1707) X_1514 = ORTHOJOIN OF X(3) (0.6818, 1.1255) X_576 = ORTHOHARMONIC OF X(187) (0.6500, 0.3498) X_1353 = REFLECTION OF X(5) IN X(6) (0.6439, 0.2855) X_515 = ISOGONAL CONJUGATE OF X(102) (0.6804, 0.7691) X_1483 = REFLECTION OF X(5) IN X(1) (0.5867, 0.2177) X_53 = SYMMEDIAN POINT OF ORTHIC TRIANGLE (0.6164, 0.4390) X_3284 = INTERSECTION OF LINES X(2)X(340) AND X(3)X(6) (0.1903, -0.4095) X_3627 = H-1(X(3); M,-1/2) (0.6684, 0.5426) A B C X2 X3 X4 X143 X185 X156 X1514 X576 X1353 X515 X1483 X53 X3284 X3627 (a, b, c) = (5, 6, 7) | AB = 1

X3627 (H⁻¹(X3; M,-1/2)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(bc[a^{2}(b^{2} + c^{2} - 4a^{2}) + 3(b^{2} - c^{2})^{2}] : :\)
Search Value (6, 9, 13): \(-11.906110563488\)
▸ Linear Combinations1 combination
▸ Incident Central Lines14 lines

{2, 3}, {53, 3284}, {143, 185}, {156, 1514}, {495, 3585}, {496, 3583}, {515, 1483}, {576, 1353}, {1173, 3521}, {1478, 3303}, {1479, 3304}, {1484, 2829}, {1493, 2883}, {1533, 3574}

▸ Functional Relations6 relations

Notes

As a point on the Euler line, X3627 has Shinagawa coefficients \((1, -7)\).

Let ABC be a triangle and Γa a variable circle passing through A and the midpoint, Ma, of side BC. Let Ab and Ac be the points, other than A and Ma, in which the circle Γa meets AC and AB, respectively. Let A' be the vertex of parallelogram with vertices A, Ab, A', Ac. When Γa varies, the locus of A' is a line ℓa. Define ℓb and ℓc cyclically. The triangle formed by the lines ℓa, ℓb, ℓc is orthologic to ABC with orthology center X3627. The reciprocal orthologic center of these triangles is X3. See X3627 (X(3627).png). (Angel Montesdeoca, July 13, 2022)