← etc / Encyclopedia of Triangle Centers X1512–X1600 · Orthojoins, Moses Circle Centers & Star-Named Centers / X_1551 ← Prev Overview · Code Next → X1551¶ ORTHOJOIN OF X111 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 { 542, 1550 }: Central Line L(542, 1550) 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_542 = DIRECTION OF VECTOR AX + BX + CX, where X = X(98) (0.1649, 0.6534) X_1550 = ORTHOJOIN OF X(110) (0.9211, 0.1083) X_1551 = ORTHOJOIN OF X(111) (0.6353, 0.4562) A B C X2 X3 X4 X542 X1550 X1551 (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 { 542, 1550 }: Central Line L(542, 1550) 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_542 = DIRECTION OF VECTOR AX + BX + CX, where X = X(98) (0.2569, 0.6567) X_1550 = ORTHOJOIN OF X(110) (0.8489, 0.2363) X_1551 = ORTHOJOIN OF X(111) (0.6622, 0.5563) A B C X2 X3 X4 X542 X1550 X1551 (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 { 542, 1550 }: Central Line L(542, 1550) 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_542 = DIRECTION OF VECTOR AX + BX + CX, where X = X(98) (0.2881, 0.6791) X_1550 = ORTHOJOIN OF X(110) (0.8090, 0.3003) X_1551 = ORTHOJOIN OF X(111) (0.6516, 0.6242) A B C X2 X3 X4 X542 X1550 X1551 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 { 542, 1550 }: Central Line L(542, 1550) 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_542 = DIRECTION OF VECTOR AX + BX + CX, where X = X(98) (0.3962, 0.7325) X_1550 = ORTHOJOIN OF X(110) (0.6931, 0.7354) X_1551 = ORTHOJOIN OF X(111) (0.6462, 0.9379) A B C X2 X3 X4 X542 X1550 X1551 Kimberling (6, 9, 13) | AB = 1 X1551 (ORTHOJOIN OF X111) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶ Constructed Elements Central Lines: {2, 3} (Euler line), {542, 1550} Related Centers: X2 (CENTROID), X3 (CIRCUMCENTER), X4 (ORTHOCENTER), X542 (DIRECTION OF VECTOR AX + BX + CX, where X = X), X1550 (ORTHOJOIN OF X110) Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) (see note above X1512) Search Value (6, 9, 13): \(-6.900723955245\) ▸ Incident Central Lines2 lines {2, 3}, {542, 1550} Notes¶ As a point on the Euler line, X1551 has Shinagawa coefficients \((2(E + F)^{3} - 9(E - 2F)S^{2}, -3(E + F)[(E +10F)(E + F) - 6S^{2}])\).
