X96
ISOGONAL CONJUGATE OF X52
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, 54 }: Central Line L(2, 54) { 76, 95 }: Central Line L(76, 95) { 4, 231 }: Central Line L(4, 231) { 24, 847 }: Central Line L(24, 847) { 94, 925 }: Central Line L(94, 925)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_52 = ORTHOCENTER OF ORTHIC TRIANGLE (0.7155, 0.4580) X_2 = CENTROID (0.5374, 0.2000) X_54 = KOSNITA POINT (0.5880, 0.2176) X_76 = 3rd BROCARD POINT (0.4567, 0.1388) X_95 = CEVAPOINT OF CENTROID AND CIRCUMCENTER (0.5172, 0.1562) X_4 = ORTHOCENTER (0.6122, 0.3958) X_231 = -CEVA CONJUGATE OF X(128) (1.0375, 0.6458) X_24 = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE (0.6627, 0.5278) X_847 = -CROSS CONJUGATE OF X(4) (-0.1806, -0.0821) X_94 = ISOGONAL CONJUGATE OF X(50) (0.9069, -0.1331) X_925 = ANTICOMPLEMENT OF X(136) (0.0074, -0.0312) X_96 = ISOGONAL CONJUGATE OF X(52) (-0.0943, -0.0197)
A
B
C
X52
X2
X54
X76
X95
X4
X231
X24
X847
X94
X925
X96
(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, 54 }: Central Line L(2, 54) { 76, 95 }: Central Line L(76, 95) { 4, 231 }: Central Line L(4, 231) { 94, 925 }: Central Line L(94, 925)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_52 = ORTHOCENTER OF ORTHIC TRIANGLE (0.5218, 0.4800) X_2 = CENTROID (0.5467, 0.1600) X_54 = KOSNITA POINT (0.6400, 0.1600) X_76 = 3rd BROCARD POINT (0.4124, 0.0899) X_95 = CEVAPOINT OF CENTROID AND CIRCUMCENTER (0.5280, 0.0960) X_4 = ORTHOCENTER (0.6400, 0.4800) X_231 = -CEVA CONJUGATE OF X(128) (0.9181, 0.3989) X_94 = ISOGONAL CONJUGATE OF X(50) (0.8532, -0.2508) X_925 = ANTICOMPLEMENT OF X(136) (0.3600, -0.4800) X_96 = ISOGONAL CONJUGATE OF X(52) (1.7371, 0.1600)
A
B
C
X52
X2
X54
X76
X95
X4
X231
X94
X925
X96
(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, 54 }: Central Line L(2, 54) { 76, 95 }: Central Line L(76, 95) { 4, 231 }: Central Line L(4, 231) { 94, 925 }: Central Line L(94, 925)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_52 = ORTHOCENTER OF ORTHIC TRIANGLE (0.9018, 0.4859) X_2 = CENTROID (0.5393, 0.1620) X_54 = KOSNITA POINT (0.6180, 0.1620) X_76 = 3rd BROCARD POINT (0.4271, 0.0928) X_95 = CEVAPOINT OF CENTROID AND CIRCUMCENTER (0.5236, 0.0972) X_4 = ORTHOCENTER (0.6180, 0.4859) X_231 = -CEVA CONJUGATE OF X(128) (0.8795, 0.4223) X_94 = ISOGONAL CONJUGATE OF X(50) (0.8111, -0.3087) X_925 = ANTICOMPLEMENT OF X(136) (0.3820, -0.4859) X_96 = ISOGONAL CONJUGATE OF X(52) (1.9514, 0.1620)
A
B
C
X52
X2
X54
X76
X95
X4
X231
X94
X925
X96
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, 54 }: Central Line L(2, 54) { 76, 95 }: Central Line L(76, 95) { 4, 231 }: Central Line L(4, 231) { 24, 847 }: Central Line L(24, 847) { 94, 925 }: Central Line L(94, 925)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_52 = ORTHOCENTER OF ORTHIC TRIANGLE (0.7045, 0.2975) X_2 = CENTROID (0.5444, 0.0934) X_54 = KOSNITA POINT (1.8800, -0.0730) X_76 = 3rd BROCARD POINT (0.3495, 0.0360) X_95 = CEVAPOINT OF CENTROID AND CIRCUMCENTER (12.6715, -0.2255) X_4 = ORTHOCENTER (0.6331, 0.8294) X_231 = -CEVA CONJUGATE OF X(128) (1.0516, 0.1599) X_24 = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE (0.5555, 0.1854) X_847 = -CROSS CONJUGATE OF X(4) (0.7261, 0.1368) X_94 = ISOGONAL CONJUGATE OF X(50) (0.6355, 0.3039) X_925 = ANTICOMPLEMENT OF X(136) (0.7747, 0.2253) X_96 = ISOGONAL CONJUGATE OF X(52) (1.1396, 0.0192)
A
B
C
X52
X2
X54
X76
X95
X4
X231
X24
X847
X94
X925
X96
Kimberling (6, 9, 13) | AB = 1
X96 (ISOGONAL CONJUGATE OF X52 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(\sec 2A \sec(B - C) : \sec 2B \sec(C - A) : \sec 2C \sec(A - B)\)
Barycentrics: \(a \sec 2A \sec(B - C) : b \sec 2B \sec(C - A) : c \sec 2C \sec(A - B)\)
Search Value (6, 9, 13): \(-1.299749991706\)
▸ Incident Central Lines5 lines
{2 54 4 231 24 847 76 95 94 925
▸ Functional Relations8 relations
Notes
Let Refa Refb Refc be the reflection triangle . Let BA and CA be the orthogonal projections of Refb and Refc on line BC , resp. Let (OA ) be the circle with segment BA CA as diameter. Define (OB ) and (OC ) cyclically. X96 OA ), (OB ), (OC ). (Randy Hutson , June 7, 2019)
