X565
INTERSECTION OF LINES X49 X_93 AND X(143,324)
Clark Kimberling and
Cyril Parry , “Products, Square Roots, and Layers in Triangle Geometry,” Mathematics and Informatics Quarterly 10 (2000) 9-22; on page 21, section 8.
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) { 49, 93 }: Central Line L(49, 93) { 143, 324 }: Central Line L(143, 324)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.6275, 0.1789) X_93 = ISOGONAL CONJUGATE OF X(49) (0.5299, 0.2878) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6358, 0.3534) X_324 = ISOTOMIC CONJUGATE OF X(97) (0.5855, 0.3796) X_565 = INTERSECTION OF LINES X(49)X(93) AND X(143,324) (0.3278, 0.5134)
A
B
C
X49
X93
X143
X324
X565
(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) { 49, 93 }: Central Line L(49, 93) { 143, 324 }: Central Line L(143, 324)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.7800, 0.0000) X_93 = ISOGONAL CONJUGATE OF X(49) (0.6400, 0.4800) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.7120, 0.2331) X_324 = ISOTOMIC CONJUGATE OF X(97) (0.6400, 0.4800) X_565 = INTERSECTION OF LINES X(49)X(93) AND X(143,324) (1.0864, -1.0506)
A
B
C
X49
X93
X143
X324
X565
(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) { 49, 93 }: Central Line L(49, 93) { 143, 324 }: Central Line L(143, 324)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.7361, 0.0000) X_93 = ISOGONAL CONJUGATE OF X(49) (0.6180, 0.4859) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6491, 0.3580) X_324 = ISOTOMIC CONJUGATE OF X(97) (0.6180, 0.4859) X_565 = INTERSECTION OF LINES X(49)X(93) AND X(143,324) (1.1475, -1.6938)
A
B
C
X49
X93
X143
X324
X565
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) { 49, 93 }: Central Line L(49, 93) { 143, 324 }: Central Line L(143, 324)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.4207, 0.3163) X_93 = ISOGONAL CONJUGATE OF X(49) (-1.9471, 0.4523) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6356, 0.2874) X_324 = ISOTOMIC CONJUGATE OF X(97) (0.7058, 0.2879) X_565 = INTERSECTION OF LINES X(49)X(93) AND X(143,324) (0.8932, 0.2891)
A
B
C
X49
X93
X143
X324
X565
Kimberling (6, 9, 13) | AB = 1
X565 (INTERSECTION OF LINES X49 X_93 AND X(143,324) ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(\cos(3B - 3C) : \cos(3C - 3A) : \cos(3A - 3B)\)
Barycentrics: \(\sin A \cos(3B - 3C) : \sin B \cos(3C - 3A) : \sin C \cos(3A - 3B)\)
Search Value (6, 9, 13): \(-2.145101227873\)
▸ Incident Central Lines2 lines
