X3658
ISOGONAL CONJUGATE OF AYME PERSPECTOR
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 { 100, 110 }: Central Line L(100, 110) { 108, 925 }: Central Line L(108, 925) { 476, 1290 }: Central Line L(476, 1290) { 1292, 1302 }: Central Line L(1292, 1302) { 109, 2617 }: Central Line L(109, 2617)
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_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_108 = Ψ(CIRCUMCENTER, INCENTER) (0.5320, 0.6114) X_925 = ANTICOMPLEMENT OF X(136) (0.0074, -0.0312) X_476 = TIXIER POINT (0.9824, 0.2685) X_1290 = COLLINGS TRANSFORM OF X(36) (1.0092, 0.1355) X_1292 = COLLINGS TRANSFORM OF X(55) (0.0286, 0.2974) X_1302 = COLLINGS TRANSFORM OF X(381) (0.3827, 0.5987) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.3636, 0.5938) X_2617 = ISOGONAL CONJUGATE OF X(2616) (0.2897, 0.5327) X_3658 = ISOGONAL CONJUGATE OF AYME PERSPECTOR (0.8378, 0.9859)
A
B
C
X2
X3
X4
X100
X110
X108
X925
X476
X1290
X1292
X1302
X109
X2617
X3658
(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 { 100, 110 }: Central Line L(100, 110) { 108, 925 }: Euler Line { 476, 1290 }: Central Line L(476, 1290) { 1292, 1302 }: Central Line L(1292, 1302) { 109, 2617 }: Central Line L(109, 2617)
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_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2000, 0.4000) X_110 = FOCUS OF KIEPERT PARABOLA (0.3600, 0.4800) X_108 = Ψ(CIRCUMCENTER, INCENTER) (0.6400, 0.4800) X_925 = ANTICOMPLEMENT OF X(136) (0.3600, -0.4800) X_476 = TIXIER POINT (0.8761, 0.3295) X_1290 = COLLINGS TRANSFORM OF X(36) (0.9680, 0.1760) X_1292 = COLLINGS TRANSFORM OF X(55) (0.0200, 0.1400) X_1302 = COLLINGS TRANSFORM OF X(381) (0.4971, 0.5000) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.5000, 0.5000) X_2617 = ISOGONAL CONJUGATE OF X(2616) (0.4394, 0.4535) X_3658 = ISOGONAL CONJUGATE OF AYME PERSPECTOR (0.6878, 0.6439)
A
B
C
X2
X3
X4
X100
X110
X108
X925
X476
X1290
X1292
X1302
X109
X2617
X3658
(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 { 100, 110 }: Central Line L(100, 110) { 108, 925 }: Euler Line { 476, 1290 }: Central Line L(476, 1290) { 1292, 1302 }: Central Line L(1292, 1302) { 109, 2617 }: Central Line L(109, 2617)
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_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2386, 0.4263) X_110 = FOCUS OF KIEPERT PARABOLA (0.3820, 0.4859) X_108 = Ψ(CIRCUMCENTER, INCENTER) (0.6180, 0.4859) X_925 = ANTICOMPLEMENT OF X(136) (0.3820, -0.4859) X_476 = TIXIER POINT (0.8278, 0.3776) X_1290 = COLLINGS TRANSFORM OF X(36) (0.9278, 0.2588) X_1292 = COLLINGS TRANSFORM OF X(55) (0.0437, 0.2045) X_1302 = COLLINGS TRANSFORM OF X(381) (0.4983, 0.5000) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.5000, 0.5000) X_2617 = ISOGONAL CONJUGATE OF X(2616) (0.4495, 0.4668) X_3658 = ISOGONAL CONJUGATE OF AYME PERSPECTOR (0.6446, 0.5951)
A
B
C
X2
X3
X4
X100
X110
X108
X925
X476
X1290
X1292
X1302
X109
X2617
X3658
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 { 100, 110 }: Central Line L(100, 110) { 108, 925 }: Central Line L(108, 925) { 476, 1290 }: Central Line L(476, 1290) { 1292, 1302 }: Central Line L(1292, 1302) { 109, 2617 }: Central Line L(109, 2617)
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_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.3206, 0.2669) X_110 = FOCUS OF KIEPERT PARABOLA (0.4975, 0.2958) X_108 = Ψ(CIRCUMCENTER, INCENTER) (0.6487, 0.2761) X_925 = ANTICOMPLEMENT OF X(136) (0.7747, 0.2253) X_476 = TIXIER POINT (0.6381, 0.2788) X_1290 = COLLINGS TRANSFORM OF X(36) (0.8030, 0.2087) X_1292 = COLLINGS TRANSFORM OF X(55) (-0.0692, -0.2359) X_1302 = COLLINGS TRANSFORM OF X(381) (0.5533, 0.2933) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.6048, 0.2861) X_2617 = ISOGONAL CONJUGATE OF X(2616) (0.6280, 0.2716) X_3658 = ISOGONAL CONJUGATE OF AYME PERSPECTOR (0.5702, 0.3077)
A
B
C
X2
X3
X4
X100
X110
X108
X925
X476
X1290
X1292
X1302
X109
X2617
X3658
Kimberling (6, 9, 13) | AB = 1
X3658 (ISOGONAL CONJUGATE OF AYME PERSPECTOR ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) , where \(f(a,b,c) = [2a(b + c)s_{B}s_{C} - S^{2} - S_{B}S_{C}]/(b^{2} - c^{2})\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(0.210455276618\)
X3658
▸ Incident Central Lines6 lines
{2 3 100 110 108 925 109 2617 476 1290 1292 1302
▸ Functional Relations1 relation
Notes
As a point on the Euler line, X3658 \((\$aS_{A}^{3}\$(E+F) -\$aS_{A}^{2}\$[(E+F)^{2}-2S^{2}] +3\$aS_{A}\$FS^{2}-\$a\$(E+F)FS^{2}, 2\$aS_{B}S_{C}\$S^{2} -2\$aS_{A}^{2}\$S^{2})\) .
