X3032
INTERSECTION X10 X_11∩X100 X_386
X3032 = inverse-in-excircles-radical-circle of
X121
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) { 10, 11 }: Central Line L(10, 11) { 100, 386 }: Central Line L(100, 386) { 104, 573 }: Central Line L(104, 573) { 214, 1015 }: Central Line L(214, 1015) { 952, 970 }: Central Line L(952, 970) { 119, 2051 }: Central Line L(119, 2051)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_386 = INVERSE-IN-BROCARD-CIRCLE OF X(58) (0.5912, 0.2526) X_104 = ANTIPODE OF X(100) (0.8571, -0.2624) X_573 = ORTHOHARMONIC OF X(58) (0.4486, 0.0172) X_214 = -CEVA CONJUGATE OF X(44) (0.3571, 0.3499) X_1015 = EXSIMILICENTER OF MOSES CIRCLE AND INCIRCLE (0.7982, 0.1348) X_952 = INTERSECTION OF X(1)X(5) AND X(3)X(8) (0.1267, 0.6191) X_970 = CENTER OF THE APOLLONIUS CIRCLE (0.2643, -0.2872) X_119 = FEUERBACH ANTIPODE (0.3776, 0.4312) X_2051 = EXSIMILICENTER(NINE-POINT CIRCLE, APOLLONIUS CIRCLE) (0.6034, 0.3358) X_3032 = INTERSECTION X(10)X(11)∩X(100)X(386) (-1.0163, 1.0198)
A
B
C
X10
X11
X100
X386
X104
X573
X214
X1015
X952
X970
X119
X2051
X3032
(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) { 10, 11 }: Central Line L(10, 11) { 100, 386 }: Central Line L(100, 386) { 104, 573 }: Central Line L(104, 573) { 214, 1015 }: Central Line L(214, 1015) { 952, 970 }: Central Line L(952, 970) { 119, 2051 }: Central Line L(119, 2051)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_10 = SPIEKER CENTER (0.5200, 0.1400) X_11 = FEUERBACH POINT (0.7200, 0.0400) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2000, 0.4000) X_386 = INVERSE-IN-BROCARD-CIRCLE OF X(58) (0.6296, 0.2222) X_104 = ANTIPODE OF X(100) (0.8000, -0.4000) X_573 = ORTHOHARMONIC OF X(58) (0.4255, -0.1277) X_214 = -CEVA CONJUGATE OF X(44) (0.4000, 0.3000) X_1015 = EXSIMILICENTER OF MOSES CIRCLE AND INCIRCLE (0.8163, 0.1224) X_952 = INTERSECTION OF X(1)X(5) AND X(3)X(8) (0.2017, 0.6200) X_970 = CENTER OF THE APOLLONIUS CIRCLE (0.1500, -0.6000) X_119 = FEUERBACH ANTIPODE (0.4200, 0.4400) X_2051 = EXSIMILICENTER(NINE-POINT CIRCLE, APOLLONIUS CIRCLE) (0.6356, 0.3713) X_3032 = INTERSECTION X(10)X(11)∩X(100)X(386) (-0.9600, 0.8800)
A
B
C
X10
X11
X100
X386
X104
X573
X214
X1015
X952
X970
X119
X2051
X3032
(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) { 10, 11 }: Central Line L(10, 11) { 100, 386 }: Central Line L(100, 386) { 104, 573 }: Central Line L(104, 573) { 214, 1015 }: Central Line L(214, 1015) { 952, 970 }: Central Line L(952, 970) { 119, 2051 }: Central Line L(119, 2051)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_10 = SPIEKER CENTER (0.5170, 0.1419) X_11 = FEUERBACH POINT (0.6897, 0.0298) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2386, 0.4263) X_386 = INVERSE-IN-BROCARD-CIRCLE OF X(58) (0.6091, 0.2246) X_104 = ANTIPODE OF X(100) (0.7614, -0.4263) X_573 = ORTHOHARMONIC OF X(58) (0.4375, -0.1285) X_214 = -CEVA CONJUGATE OF X(44) (0.4114, 0.3142) X_1015 = EXSIMILICENTER OF MOSES CIRCLE AND INCIRCLE (0.7920, 0.1010) X_952 = INTERSECTION OF X(1)X(5) AND X(3)X(8) (0.2388, 0.6521) X_970 = CENTER OF THE APOLLONIUS CIRCLE (0.2080, -0.6010) X_119 = FEUERBACH ANTIPODE (0.4283, 0.4561) X_2051 = EXSIMILICENTER(NINE-POINT CIRCLE, APOLLONIUS CIRCLE) (0.6143, 0.3758) X_3032 = INTERSECTION X(10)X(11)∩X(100)X(386) (-0.7528, 0.9660)
A
B
C
X10
X11
X100
X386
X104
X573
X214
X1015
X952
X970
X119
X2051
X3032
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) { 10, 11 }: Central Line L(10, 11) { 100, 386 }: Central Line L(100, 386) { 104, 573 }: Central Line L(104, 573) { 214, 1015 }: Central Line L(214, 1015) { 952, 970 }: Central Line L(952, 970) { 119, 2051 }: Central Line L(119, 2051)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_10 = SPIEKER CENTER (0.5089, 0.0750) X_11 = FEUERBACH POINT (0.6563, 0.0066) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.3206, 0.2669) X_386 = INVERSE-IN-BROCARD-CIRCLE OF X(58) (0.6513, 0.1486) X_104 = ANTIPODE OF X(100) (0.6794, -0.8162) X_573 = ORTHOHARMONIC OF X(58) (0.4096, -0.5275) X_214 = -CEVA CONJUGATE OF X(44) (0.4680, 0.1984) X_1015 = EXSIMILICENTER OF MOSES CIRCLE AND INCIRCLE (0.8131, 0.0702) X_952 = INTERSECTION OF X(1)X(5) AND X(3)X(8) (0.3380, 0.7161) X_970 = CENTER OF THE APOLLONIUS CIRCLE (0.0513, -1.5299) X_119 = FEUERBACH ANTIPODE (0.4769, 0.5481) X_2051 = EXSIMILICENTER(NINE-POINT CIRCLE, APOLLONIUS CIRCLE) (0.6409, 0.5381) X_3032 = INTERSECTION X(10)X(11)∩X(100)X(386) (-0.6603, 0.6177)
A
B
C
X10
X11
X100
X386
X104
X573
X214
X1015
X952
X970
X119
X2051
X3032
Kimberling (6, 9, 13) | AB = 1
X3032 (INTERSECTION X10 X_11∩X100 X_386 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \((b^{3} + b^{2}c - ba^{2} + abc - a^{2}c - ac^{2})(c^{3} + c^{2}b - ca^{2} + abc - a^{2}b - ab^{2}) : :\)
Search Value (6, 9, 13): \(6.713846098455\)
X3032
▸ Incident Central Lines9 lines
{10 11 100 386 104 573 119 2051 214 1015 528 3034 952 970 1695 1768 2787 3029
▸ Functional Relations2 relations
Notes
Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC , and define B and C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the excentral triangle . X3032 X74 A'B'C' . (Randy Hutson , August 29, 2018)
