X2043

3rd EULER-VECTEN-GIBERT POINT

X2043 = reflection of X2044 in X3
< 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) { 2044, 3 }: Euler Line { 2, 3 }: Euler Line { 13, 485 }: Central Line L(13, 485) { 14, 486 }: Central Line L(14, 486) { 298, 637 }: Central Line L(298, 637) { 299, 638 }: Central Line L(299, 638) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2044 = 4th EULER-VECTEN-GIBERT POINT (0.5648, 0.2716) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_2 = CENTROID (0.5374, 0.2000) X_13 = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT) (0.5879, 0.2819) X_485 = VECTEN POINT (0.5773, 0.2578) X_14 = 2nd ISOGONIC CENTER (0.9134, -0.1143) X_486 = INNER VECTEN POINT (-0.0810, -0.0170) X_298 = ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER (0.4724, 0.1301) X_637 = ANTICOMPLEMENT OF X(371) (0.4819, 0.1805) X_299 = ISOTOMIC CONJUGATE OF 2nd ISOGONIC CENTER (-1.1465, -1.6097) X_638 = ANTICOMPLEMENT OF X(372) (0.1827, -0.3137) X_2043 = 3rd EULER-VECTEN-GIBERT POINT (0.4352, -0.0675) A B C X2044 X3 X2 X13 X485 X14 X486 X298 X637 X299 X638 X2043 (a, b, c) = (5, 6, 7) | AB = 1

X2043 (3rd EULER-VECTEN-GIBERT POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc[- 2(b^{2} - c^{2})^{2} + a^{2}(a^{2} + b^{2} + c^{2}) + 3^{1/2}[(b^{2} - c^{2})^{2} - a^{2}(b^{2} + c^{2})]]\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(13.975559681888\)
▸ Linear Combinations1 combination
▸ Incident Central Lines11 lines

{2, 3}, {13, 485}, {14, 486}, {17, 1327}, {18, 1328}, {298, 637}, {299, 638}, {489, 616}, {490, 617}, {491, 622}, {492, 621}

▸ Functional Relations7 relations

Notes

As a point on the Euler line, X2043 has Shinagawa coefficients \((3^{1/2} - 1, -3 + 3^{1/2})\).