← etc / Encyclopedia of Triangle Centers X1512–X1600 · Orthojoins, Moses Circle Centers & Star-Named Centers / X_1536 ← Prev Overview · Code Next → X1536¶ ORTHOJOIN OF X42 X1536 = reflection of X1530 in X1541 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) { 1530, 1541 }: Central Line L(1530, 1541) { 2, 3 }: Euler Line { 118, 516 }: Central Line L(118, 516) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1530 = ORTHOJOIN OF X(31) (-0.4898, -1.6288) X_1541 = ORTHOJOIN OF X(55) (0.9271, 1.6491) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_118 = MIDPOINT OF X(4) AND X(101) (0.4133, 0.4603) X_516 = ISOGONAL CONJUGATE OF X(103) (0.7703, 0.7386) X_1536 = ORTHOJOIN OF X(42) (2.3440, 4.9269) A B C X1530 X1541 X2 X3 X118 X516 X1536 (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) { 1530, 1541 }: Central Line L(1530, 1541) { 2, 3 }: Euler Line { 118, 516 }: Central Line L(118, 516) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_1530 = ORTHOJOIN OF X(31) (0.0862, -0.6277) X_1541 = ORTHOJOIN OF X(55) (0.7508, 1.2554) X_2 = CENTROID (0.5467, 0.1600) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_118 = MIDPOINT OF X(4) AND X(101) (0.4738, 0.4708) X_516 = ISOGONAL CONJUGATE OF X(103) (0.7380, 0.7022) X_1536 = ORTHOJOIN OF X(42) (1.4154, 3.1385) A B C X1530 X1541 X2 X3 X118 X516 X1536 (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) { 1530, 1541 }: Central Line L(1530, 1541) { 2, 3 }: Euler Line { 118, 516 }: Central Line L(118, 516) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_1530 = ORTHOJOIN OF X(31) (0.1455, -0.6501) X_1541 = ORTHOJOIN OF X(55) (0.7135, 1.2835) X_2 = CENTROID (0.5393, 0.1620) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_118 = MIDPOINT OF X(4) AND X(101) (0.4773, 0.4792) X_516 = ISOGONAL CONJUGATE OF X(103) (0.7014, 0.7136) X_1536 = ORTHOJOIN OF X(42) (1.2815, 3.2170) A B C X1530 X1541 X2 X3 X118 X516 X1536 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) { 1530, 1541 }: Central Line L(1530, 1541) { 2, 3 }: Euler Line { 118, 516 }: Central Line L(118, 516) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_1530 = ORTHOJOIN OF X(31) (0.5187, 0.4279) X_1541 = ORTHOJOIN OF X(55) (0.6419, 1.1763) X_2 = CENTROID (0.5444, 0.0934) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_118 = MIDPOINT OF X(4) AND X(101) (0.5407, 0.5614) X_516 = ISOGONAL CONJUGATE OF X(103) (0.6510, 0.7407) X_1536 = ORTHOJOIN OF X(42) (0.7652, 1.9247) A B C X1530 X1541 X2 X3 X118 X516 X1536 Kimberling (6, 9, 13) | AB = 1 X1536 (ORTHOJOIN OF X42) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶ Constructed Elements Central Lines: {1530, 1541}, {2, 3} (Euler line) Related Centers: X1530 (ORTHOJOIN OF X31), X1541 (ORTHOJOIN OF X55), X2 (CENTROID), X3 (CIRCUMCENTER) Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) (see note above X1512) Search Value (6, 9, 13): \(-18.036282990854\) ▸ Incident Central Lines2 lines {2, 3}, {118, 516} ▸ Functional Relations1 relation X1536 = reflection of X1530 in X1541 Notes¶ As a point on the Euler line, X1536 has Shinagawa coefficients \((S^{2}, -2(E + F)^{2} + 2\$bcS_{A}\$ - S^{2})\).
