← etc / Encyclopedia of Triangle Centers X1512–X1600 · Orthojoins, Moses Circle Centers & Star-Named Centers / X_1542 ← Prev Overview · Code Next → X1542¶ ORTHOJOIN OF X56 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) { 4, 9 }: Orthocenter–Spieker Line { 117, 515 }: Central Line L(117, 515) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_4 = ORTHOCENTER (0.6122, 0.3958) X_9 = MITTENPUNKT (0.5110, 0.1615) X_117 = MIDPOINT OF X(4) AND X(109) (0.4879, 0.4948) X_515 = ISOGONAL CONJUGATE OF X(102) (0.6804, 0.7691) X_1542 = ORTHOJOIN OF X(56) (0.2560, -0.4283) A B C X4 X9 X117 X515 X1542 (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) { 4, 9 }: Orthocenter–Spieker Line { 117, 515 }: Central Line L(117, 515) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_4 = ORTHOCENTER (0.6400, 0.4800) X_9 = MITTENPUNKT (0.5091, 0.1091) X_117 = MIDPOINT OF X(4) AND X(109) (0.5700, 0.4900) X_515 = ISOGONAL CONJUGATE OF X(102) (0.6280, 0.7292) X_1542 = ORTHOJOIN OF X(56) (0.5200, 0.1400) A B C X4 X9 X117 X515 X1542 (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) { 4, 9 }: Orthocenter–Spieker Line { 117, 515 }: Central Line L(117, 515) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_4 = ORTHOCENTER (0.6180, 0.4859) X_9 = MITTENPUNKT (0.5077, 0.1103) X_117 = MIDPOINT OF X(4) AND X(109) (0.5590, 0.4929) X_515 = ISOGONAL CONJUGATE OF X(102) (0.6077, 0.7329) X_1542 = ORTHOJOIN OF X(56) (0.5170, 0.1419) A B C X4 X9 X117 X515 X1542 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) { 4, 9 }: Orthocenter–Spieker Line { 117, 515 }: Central Line L(117, 515) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_4 = ORTHOCENTER (0.6331, 0.8294) X_9 = MITTENPUNKT (0.5022, 0.0343) X_117 = MIDPOINT OF X(4) AND X(109) (0.6190, 0.5577) X_515 = ISOGONAL CONJUGATE OF X(102) (0.5610, 0.7492) X_1542 = ORTHOJOIN OF X(56) (0.6246, 0.7773) A B C X4 X9 X117 X515 X1542 Kimberling (6, 9, 13) | AB = 1 X1542 (ORTHOJOIN OF X56) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶ Constructed Elements Central Lines: {4, 9}, {117, 515} Related Centers: X4 (ORTHOCENTER), X9 (MITTENPUNKT), X117 (MIDPOINT OF X4 AND X109), X515 (ISOGONAL CONJUGATE OF X102) Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) (see note above X1512) Search Value (6, 9, 13): \(-5.070379072769\) ▸ Incident Central Lines2 lines {4, 9}, {117, 515}
