X3109

ORTHOGONAL PROJECTION OF X1 ON EULER LINE

X3109 = orthogonal projection of X1 on Euler line
< 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 { 1, 523 }: Central Line L(1, 523) { 11, 759 }: Central Line L(11, 759) { 110, 952 }: Central Line L(110, 952) { 163, 1146 }: Central Line L(163, 1146) { 476, 953 }: Central Line L(476, 953) 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_1 = INCENTER (0.5714, 0.2333) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0856, -0.0096) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_759 = EVEN (1, 3) CIRCUMCIRCLE POINT (0.9498, -0.1389) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_952 = INTERSECTION OF X(1)X(5) AND X(3)X(8) (0.1267, 0.6191) X_163 = TRILINEAR PRODUCT X(6)*X(110) (0.3174, 0.5883) X_1146 = CENTER OF HYPERBOLA {{A,B,C,X(2),X(8)}} (0.6866, 0.0428) X_476 = TIXIER POINT (0.9824, 0.2685) X_953 = ISOGONAL CONJUGATE OF X(952) (0.0000, 0.2041) X_3109 = ORTHOGONAL PROJECTION OF X(1) ON EULER LINE (0.5529, 0.2404) A B C X2 X3 X1 X523 X11 X759 X110 X952 X163 X1146 X476 X953 X3109 (a, b, c) = (5, 6, 7) | AB = 1

X3109 (ORTHOGONAL PROJECTION OF X1 ON EULER LINE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Barycentrics: \((a + b) \cdot (a + c) \cdot (2 a^{5} - 2 a^{4} b + a^{2} b^{3} - 2 a b^{4} + b^{5} - 2 a^{4} c + 2 a^{3} b c - a^{2} b^{2} c + b^{4} c - a^{2} b c^{2} + 4 a b^{2} c^{2} - 2 b^{3} c^{2} + a^{2} c^{3} - 2 b^{2} c^{3} - 2 a c^{4} + b c^{4} + c^{5}) : :\)
Search Value (6, 9, 13): \(2.126221797603\)
▸ Linear Combinations1 combination

X3109 lies on the cubic K165.

▸ Incident Central Lines36 lines

{1, 523}, {2, 3}, {11, 759}, {12, 2222}, {58, 6788}, {110, 952}, {119, 1793}, {163, 1146}, {229, 18990}, {355, 3233}, {476, 953}, {513, 13868}, {525, 24347}, {643, 1145}, {662, 10609}, {691, 2726}, {1043, 6790}, {1086, 30927}, {1290, 5253}, {1304, 2734}, {1483, 14611}, {2150, 21933}, {2617, 34586}, {3833, 34583}, {5690, 35193}, {5972, 6739}, {6789, 25079}, {10543, 15792}, {10950, 17104}, {11657, 13408}, {11809, 16332}, {12019, 24624}, {12690, 19642}, {18180, 31849}, {18653, 28160}, {25533, 34123}

▸ Functional Relations10 relations

  • X3109 = orthogonal projection of X1 on Euler line

  • X3109 = circumcircle-inverse of X859

  • X3109 = Conway-circle-inverse of X5214

  • X3109 = antigonal conjugate of X38950

  • X3109 = polar-circle-inverse of X860

  • X3109 = orthoptic-circle-of-Steiner-inellipse-inverse of X8229

  • X3109 = intersection, other than X3, of the Euler line and circle O(1,3)

  • X3109 = X110-of-X1-Brocard triangle

  • X3109 = crossdifference of every pair of points on line {647, 2245}

  • X3109 = {Xi,Xj}-harmonic conjugate of Xk for these (i,j,k): {21, 29, 14010}, {21, 3658, 3}, {29, 7452, 4}, {423, 448, 1375}, {1113, 1114, 859}, {7424, 7478, 1325}, {11101, 13746, 5}