X2787
ISOGONAL CONJUGATE OF X2703
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) { 11, 115 }: Central Line L(11, 115) { 99, 100 }: Central Line L(99, 100) { 98, 104 }: Central Line L(98, 104) { 114, 119 }: Central Line L(114, 119) { 148, 149 }: Central Line L(148, 149) { 147, 153 }: Central Line L(147, 153)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2703 = SR(X(98), X(104)) (0.9885, -0.0455) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7507, 0.0838) X_99 = STEINER POINT (0.1109, 0.4322) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_98 = TARRY POINT (0.8891, -0.2281) X_104 = ANTIPODE OF X(100) (0.8571, -0.2624) X_114 = KIEPERT ANTIPODE (0.3616, 0.4140) X_119 = FEUERBACH ANTIPODE (0.3776, 0.4312) X_148 = STEINER POINT OF ANTICOMPLEMENTARY TRIANGLE (1.3904, -0.2646) X_149 = REFLECTION OF X(20) IN X(104) (1.3265, -0.3333) X_147 = TARRY POINT OF ANTICOMPLEMENTARY TRIANGLE (-0.1660, 1.0561) X_153 = REFLECTION OF X(20) IN X(100) (-0.1020, 1.1248) X_2787 = ISOGONAL CONJUGATE OF X(2703) (0.1375, -0.2296)
A
B
C
X2703
X11
X115
X99
X100
X98
X104
X114
X119
X148
X149
X147
X153
X2787
(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) { 11, 115 }: Central Line L(11, 115) { 99, 100 }: Central Line L(99, 100) { 98, 104 }: Central Line L(98, 104) { 114, 119 }: Central Line L(114, 119) { 148, 149 }: Central Line L(148, 149) { 147, 153 }: Central Line L(147, 153)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_2703 = SR(X(98), X(104)) (0.9959, -0.0642) X_11 = FEUERBACH POINT (0.7200, 0.0400) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7445, 0.0609) X_99 = STEINER POINT (0.1511, 0.3581) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2000, 0.4000) X_98 = TARRY POINT (0.8489, -0.3581) X_104 = ANTIPODE OF X(100) (0.8000, -0.4000) X_114 = KIEPERT ANTIPODE (0.3955, 0.4191) X_119 = FEUERBACH ANTIPODE (0.4200, 0.4400) X_148 = STEINER POINT OF ANTICOMPLEMENTARY TRIANGLE (1.3378, -0.2363) X_149 = REFLECTION OF X(20) IN X(104) (1.2400, -0.3200) X_147 = TARRY POINT OF ANTICOMPLEMENTARY TRIANGLE (-0.0578, 1.1963) X_153 = REFLECTION OF X(20) IN X(100) (0.0400, 1.2800) X_2787 = ISOGONAL CONJUGATE OF X(2703) (0.1098, -0.2139)
A
B
C
X2703
X11
X115
X99
X100
X98
X104
X114
X119
X148
X149
X147
X153
X2787
(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) { 11, 115 }: Central Line L(11, 115) { 99, 100 }: Central Line L(99, 100) { 98, 104 }: Central Line L(98, 104) { 114, 119 }: Central Line L(114, 119) { 148, 149 }: Central Line L(148, 149) { 147, 153 }: Central Line L(147, 153)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_2703 = SR(X(98), X(104)) (0.9966, 0.0580) X_11 = FEUERBACH POINT (0.6897, 0.0298) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7135, 0.0464) X_99 = STEINER POINT (0.1910, 0.3931) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2386, 0.4263) X_98 = TARRY POINT (0.8090, -0.3931) X_104 = ANTIPODE OF X(100) (0.7614, -0.4263) X_114 = KIEPERT ANTIPODE (0.4045, 0.4395) X_119 = FEUERBACH ANTIPODE (0.4283, 0.4561) X_148 = STEINER POINT OF ANTICOMPLEMENTARY TRIANGLE (1.2361, -0.3003) X_149 = REFLECTION OF X(20) IN X(104) (1.1407, -0.3666) X_147 = TARRY POINT OF ANTICOMPLEMENTARY TRIANGLE (0.0000, 1.2720) X_153 = REFLECTION OF X(20) IN X(100) (0.0953, 1.3384) X_2787 = ISOGONAL CONJUGATE OF X(2703) (0.0674, -0.1666)
A
B
C
X2703
X11
X115
X99
X100
X98
X104
X114
X119
X148
X149
X147
X153
X2787
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) { 11, 115 }: Central Line L(11, 115) { 99, 100 }: Central Line L(99, 100) { 98, 104 }: Central Line L(98, 104) { 114, 119 }: Central Line L(114, 119) { 148, 149 }: Central Line L(148, 149) { 147, 153 }: Central Line L(147, 153)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_2703 = SR(X(98), X(104)) (0.9821, 0.0304) X_11 = FEUERBACH POINT (0.6563, 0.0066) X_115 = CENTER OF KIEPERT HYPERBOLA (0.6909, 0.0207) X_99 = STEINER POINT (0.2513, 0.2387) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.3206, 0.2669) X_98 = TARRY POINT (0.7487, -0.7881) X_104 = ANTIPODE OF X(100) (0.6794, -0.8162) X_114 = KIEPERT ANTIPODE (0.4422, 0.5341) X_119 = FEUERBACH ANTIPODE (0.4769, 0.5481) X_148 = STEINER POINT OF ANTICOMPLEMENTARY TRIANGLE (1.1305, -0.1974) X_149 = REFLECTION OF X(20) IN X(104) (0.9920, -0.2537) X_147 = TARRY POINT OF ANTICOMPLEMENTARY TRIANGLE (0.1358, 1.8562) X_153 = REFLECTION OF X(20) IN X(100) (0.2743, 1.9125) X_2787 = ISOGONAL CONJUGATE OF X(2703) (-0.0635, -0.1533)
A
B
C
X2703
X11
X115
X99
X100
X98
X104
X114
X119
X148
X149
X147
X153
X2787
Kimberling (6, 9, 13) | AB = 1
X2787 (ISOGONAL CONJUGATE OF X2703 ) 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 \(f(a,b,c) = bc(b - c)(a^{3} - b^{2}c - bc^{2} + abc)\) (M. Iliev, 5/13/07)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
▸ Incident Central Lines9 lines
{11 115 30 511 80 291 98 104 99 100 114 119 147 153 148 149 1019 2533
▸ Functional Relations6 relations
Notes
Let Ia be the reflection of X1 BC , and define Ib and Ic cyclically; then X2787 X512 Ia Ib Ic ; Ia Ib Ic is the inner Garcia triangle. (Randy Hutson , February 10, 2016)
