← etc / Encyclopedia of Triangle Centers X1512–X1600 · Orthojoins, Moses Circle Centers & Star-Named Centers / X_1529 ← Prev Overview · Code Next → X1529¶ ORTHOJOIN OF X25 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 { 132, 1503 }: Central Line L(132, 1503) 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_132 = ∩X(4)X(32) (0.5642, 0.5039) X_1503 = POINT ARKAB (0.4818, -0.3843) X_1529 = ORTHOJOIN OF X(25) (0.5346, 0.1925) A B C X2 X3 X4 X132 X1503 X1529 (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 { 132, 1503 }: Central Line L(132, 1503) 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_132 = ∩X(4)X(32) (0.6400, 0.4800) X_1503 = POINT ARKAB (0.5467, -0.4150) X_1529 = ORTHOJOIN OF X(25) (0.6400, 0.4800) A B C X2 X3 X4 X132 X1503 X1529 (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 { 132, 1503 }: Central Line L(132, 1503) 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_132 = ∩X(4)X(32) (0.6180, 0.4859) X_1503 = POINT ARKAB (0.5393, -0.4130) X_1529 = ORTHOJOIN OF X(25) (0.6180, 0.4859) A B C X2 X3 X4 X132 X1503 X1529 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 { 132, 1503 }: Central Line L(132, 1503) 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_132 = ∩X(4)X(32) (0.6532, 0.5491) X_1503 = POINT ARKAB (0.5683, -0.5623) X_1529 = ORTHOJOIN OF X(25) (0.6407, 0.8918) A B C X2 X3 X4 X132 X1503 X1529 Kimberling (6, 9, 13) | AB = 1 X1529 (ORTHOJOIN OF X25) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶ Constructed Elements Central Lines: {2, 3} (Euler line), {132, 1503} Related Centers: X2 (CENTROID), X3 (CIRCUMCENTER), X4 (ORTHOCENTER), X132 (∩X4X32), X1503 (POINT ARKAB) Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) (see note above X1512) Search Value (6, 9, 13): \(-6.381050587227\) ▸ Incident Central Lines2 lines {2, 3}, {132, 1503} Notes¶ As a point on the Euler line, X1529 has Shinagawa coefficients \((FS^{2}, -(E + F)[4(E + F)F - S^{2}])\).
