X1129
ISOGONAL CONJUGATE OF X1127
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) { 1, 168 }: Central Line L(1, 168)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1127 = 1st DE VILLIERS POINT (0.5779, 0.2467) X_1 = INCENTER (0.5714, 0.2333) X_168 = MITTENPUNKT OF EXCENTRAL TRIANGLE (0.4565, 0.0098) X_1129 = ISOGONAL CONJUGATE OF X(1127) (-1.2689, -1.9056)
A
B
C
X1127
X1
X168
X1129
(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) { 1, 168 }: Central Line L(1, 168)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_1127 = 1st DE VILLIERS POINT (0.6078, 0.2189) X_1 = INCENTER (0.6000, 0.2000) X_168 = MITTENPUNKT OF EXCENTRAL TRIANGLE (0.4413, -0.1462) X_1129 = ISOGONAL CONJUGATE OF X(1127) (-0.2109, -0.5921)
A
B
C
X1127
X1
X168
X1129
(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) { 1, 168 }: Central Line L(1, 168)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_1127 = 1st DE VILLIERS POINT (0.5907, 0.2214) X_1 = INCENTER (0.5841, 0.2021) X_168 = MITTENPUNKT OF EXCENTRAL TRIANGLE (0.4506, -0.1478) X_1129 = ISOGONAL CONJUGATE OF X(1127) (-0.0716, -0.5684)
A
B
C
X1127
X1
X168
X1129
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) { 1, 168 }: Central Line L(1, 168)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_1127 = 1st DE VILLIERS POINT (0.6207, 0.1558) X_1 = INCENTER (0.6154, 0.1300) X_168 = MITTENPUNKT OF EXCENTRAL TRIANGLE (0.4361, -0.5998) X_1129 = ISOGONAL CONJUGATE OF X(1127) (0.4817, -0.0045)
A
B
C
X1127
X1
X168
X1129
Kimberling (6, 9, 13) | AB = 1
X1129 (ISOGONAL CONJUGATE OF X1127 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \((\sin 3A/4)/\sin(A/4) : :\)
Search Value (6, 9, 13): \(4.134764790364\)
▸ Alternate Coordinates1 form
▸ Incident Central Lines1 line
▸ Functional Relations3 relations
X1129 X1127
X1129
X1129 ABC and cross-triangle of ABC and BCI triangle
Notes
Let A' , B' , C' be as at X1127 A″ = BC' ∩ CB' , B″ = CA' ∩ AC' , C″ = AB' ∩ BA' . The lines AA″ , BB″ , CC″ concur in X1129 A'B'C' and A″B″C″ are analogous to the 1st Morley triangle and adjunct Morley triangle, substituting angle quadrisectors for angle trisectors. (Randy Hutson , January 29, 2018)
