← etc / Encyclopedia of Triangle Centers X1512–X1600 · Orthojoins, Moses Circle Centers & Star-Named Centers / X_1559 ← Prev Overview · Code Next → X1559¶ ORTHOJOIN OF X393 X1559 = inverse-in-circumconic-centered-at-X4 of X20 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 { 133, 1515 }: Central Line L(133, 1515) 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_133 = INTERSECTION OF LINES X(4)X(74) AND X(5)X(122) (0.5928, 0.5014) X_1515 = ORTHOJOIN OF X(4) (0.5725, -0.0213) X_1559 = ORTHOJOIN OF X(393) (0.5860, 0.3272) A B C X2 X3 X4 X133 X1515 X1559 (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_1559 = ORTHOJOIN OF X(393) (0.6400, 0.4800) A B C X2 X3 X4 X1559 (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_1559 = ORTHOJOIN OF X(393) (0.6180, 0.4859) A B C X2 X3 X4 X1559 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 { 133, 1515 }: Central Line L(133, 1515) 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_133 = INTERSECTION OF LINES X(4)X(74) AND X(5)X(122) (0.6120, 0.5590) X_1515 = ORTHOJOIN OF X(4) (0.6278, 0.9761) X_1559 = ORTHOJOIN OF X(393) (0.6173, 0.6980) A B C X2 X3 X4 X133 X1515 X1559 Kimberling (6, 9, 13) | AB = 1 X1559 (ORTHOJOIN OF X393) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶ Constructed Elements Central Lines: {2, 3} (Euler line), {133, 1515} Related Centers: X2 (CENTROID), X3 (CIRCUMCENTER), X4 (ORTHOCENTER), X133 (INTERSECTION OF LINES X), X1515 (ORTHOJOIN OF X4) Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) (see note above X1512) Search Value (6, 9, 13): \(-4.193953139517\) ▸ Incident Central Lines2 lines {2, 3}, {133, 1515} ▸ Functional Relations1 relation X1559 = inverse-in-circumconic-centered-at-X4 of X20 Notes¶ As a point on the Euler line, X1559 has Shinagawa coefficients \((F^{2}, -2(2E - F)F + S^{2})\).
