← etc / Encyclopedia of Triangle Centers X1512–X1600 · Orthojoins, Moses Circle Centers & Star-Named Centers / X_1563 ← Prev Overview · Code Next → X1563¶ ORTHOJOIN OF X588 5, 6, 7 3, 4, 5 Kepler Kimberling < 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 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_1563 = ORTHOJOIN OF X(588) (1.7786, 3.4475) A B C X2 X3 X4 X1563 (a, b, c) = (5, 6, 7) | AB = 1 < 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 A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_2 = CENTROID (0.5467, 0.1600) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_4 = ORTHOCENTER (0.6400, 0.4800) X_1563 = ORTHOJOIN OF X(588) (1.1516, 2.2340) A B C X2 X3 X4 X1563 (a, b, c) = (3, 4, 5) | AB = 1 < 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 A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_2 = CENTROID (0.5393, 0.1620) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_4 = ORTHOCENTER (0.6180, 0.4859) X_1563 = ORTHOJOIN OF X(588) (1.0609, 2.3088) A B C X2 X3 X4 X1563 Kepler (1, √φ, φ) | AB = 1 < 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 A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_2 = CENTROID (0.5444, 0.0934) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_4 = ORTHOCENTER (0.6331, 0.8294) X_1563 = ORTHOJOIN OF X(588) (0.7146, 1.5047) A B C X2 X3 X4 X1563 Kimberling (6, 9, 13) | AB = 1 X1563 (ORTHOJOIN OF X588) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶ Constructed Elements Central Lines: {2, 3} (Euler line) Related Centers: X2 (CENTROID), X3 (CIRCUMCENTER), X4 (ORTHOCENTER) Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) (see note above X1512) Search Value (6, 9, 13): \(-13.296517752190\) ▸ Incident Central Lines1 line {2, 3} Notes¶ As a point on the Euler line, X1563 has Shinagawa coefficients \(([2(E + F)^{2} + (9E + 8F)S+10S^{2}]S, -(E + F)^{2}E - 2(5E - F)(E + F)S - 4(7E - 2F)S^{2} - 22S^{3})\).
