← etc / Encyclopedia of Triangle Centers X1908–X2000 · Bicentric Pairs PU(k) & Orthocorrespondents / X_1938 ← Prev Overview · Code Next → X1938¶ IDEAL POINT OF PU(15) As the isogonal conjugate of a point on the circumcircle, X1938 lies on the line at infinity. 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) { 30, 511 }: Central Line L(30, 511) { 65, 650 }: Central Line L(65, 650) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2335, -0.3020) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6041, 0.2933) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (13.7143, -4.1991) X_1938 = IDEAL POINT OF PU(15) (-0.0178, 0.3902) A B C X30 X511 X65 X650 X1938 (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) { 30, 511 }: Central Line L(30, 511) { 65, 650 }: Central Line L(65, 650) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_30 = EULER INFINITY POINT (0.7077, 0.7120) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2569, -0.3367) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6400, 0.2800) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (6.4000, -1.2000) X_1938 = IDEAL POINT OF PU(15) (-0.0102, 0.3031) A B C X30 X511 X65 X650 X1938 (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) { 30, 511 }: Central Line L(30, 511) { 65, 650 }: Central Line L(65, 650) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_30 = EULER INFINITY POINT (0.6751, 0.7207) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2881, -0.3552) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6180, 0.2838) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (7.5663, -1.2021) X_1938 = IDEAL POINT OF PU(15) (-0.0229, 0.2822) A B C X30 X511 X65 X650 X1938 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) { 30, 511 }: Central Line L(30, 511) { 65, 650 }: Central Line L(65, 650) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_30 = EULER INFINITY POINT (0.6229, 0.7447) X_511 = ISOGONAL CONJUGATE OF X(98) (0.3235, -0.5244) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6417, 0.2223) X_650 = CROSSDIFFERENCE OF X(1) AND X(3) (3.4560, -0.1300) X_1938 = IDEAL POINT OF PU(15) (-0.1066, 0.1748) A B C X30 X511 X65 X650 X1938 Kimberling (6, 9, 13) | AB = 1 X1938 (IDEAL POINT OF PU(15)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶ Constructed Elements Central Lines: {30, 511}, {65, 650} Related Centers: X30 (EULER INFINITY POINT), X511 (ISOGONAL CONJUGATE OF X98), X65 (ORTHOCENTER OF THE INTOUCH TRIANGLE), X650 (CROSSDIFFERENCE OF X1 AND X3) Trilinears: \((\cos A \cos B - \cos^{2} C) \sin B - (\cos A \cos C - \cos^{2} B) \sin C\) ▸ Incident Central Lines2 lines {30, 511}, {65, 650} ▸ Functional Relations2 relations X1938 = crossdifference of every pair of points on line X6X1936 X1938 = ideal point of PU(i) for these i: 15, 110
