X252
ISOGONAL CONJUGATE OF X143
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) { 54, 140 }: Central Line L(54, 140) { 93, 186 }: Central Line L(93, 186) { 3, 930 }: Central Line L(3, 930)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6358, 0.3534) X_54 = KOSNITA POINT (0.5880, 0.2176) X_140 = MIDPOINT OF X(3) AND X(5) (0.5281, 0.1755) X_93 = ISOGONAL CONJUGATE OF X(49) (0.5299, 0.2878) X_186 = INVERSE-IN-CIRCUMCIRCLE OF X(4) (0.7957, 0.8758) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_930 = ANTICOMPLEMENT OF X(137) (0.0497, 0.3422) X_252 = ISOGONAL CONJUGATE OF X(143) (0.4565, 0.1253)
A
B
C
X143
X54
X140
X93
X186
X3
X930
X252
(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) { 54, 140 }: Central Line L(54, 140) { 3, 930 }: Central Line L(3, 930)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.7120, 0.2331) X_54 = KOSNITA POINT (0.6400, 0.1600) X_140 = MIDPOINT OF X(3) AND X(5) (0.5350, 0.1200) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_930 = ANTICOMPLEMENT OF X(137) (0.0428, 0.2025) X_252 = ISOGONAL CONJUGATE OF X(143) (0.3705, 0.0573)
A
B
C
X143
X54
X140
X3
X930
X252
(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) { 54, 140 }: Central Line L(54, 140) { 3, 930 }: Central Line L(3, 930)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6491, 0.3580) X_54 = KOSNITA POINT (0.6180, 0.1620) X_140 = MIDPOINT OF X(3) AND X(5) (0.5295, 0.1215) X_3 = CIRCUMCENTER (0.5000, 0.0000) X_930 = ANTICOMPLEMENT OF X(137) (0.0736, 0.2611) X_252 = ISOGONAL CONJUGATE OF X(143) (0.3991, 0.0618)
A
B
C
X143
X54
X140
X3
X930
X252
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) { 54, 140 }: Central Line L(54, 140) { 93, 186 }: Central Line L(93, 186) { 3, 930 }: Central Line L(3, 930)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6356, 0.2874) X_54 = KOSNITA POINT (1.8800, -0.0730) X_140 = MIDPOINT OF X(3) AND X(5) (0.5333, 0.0013) X_93 = ISOGONAL CONJUGATE OF X(49) (-1.9471, 0.4523) X_186 = INVERSE-IN-CIRCUMCIRCLE OF X(4) (0.5350, 0.0159) X_3 = CIRCUMCENTER (0.5000, -0.2747) X_930 = ANTICOMPLEMENT OF X(137) (0.2137, -0.7681) X_252 = ISOGONAL CONJUGATE OF X(143) (0.6562, -0.0054)
A
B
C
X143
X54
X140
X93
X186
X3
X930
X252
Kimberling (6, 9, 13) | AB = 1
X252 (ISOGONAL CONJUGATE OF X143 ) 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 \(f(A,B,C) = \sec(B - C)/[1 - 2 \cos(2A)]\)
Barycentrics: (sin A)f(
A ,
B ,
C ) : (sin B)f(
B ,
C ,
A ) : (sin C) f(
C ,
A ,
B ) Let
Ha Hb Hc be the
orthic triangle . Let
OA be the circle with center A passing through
Ha . Let
A″ be the intersection, other than
Ha , of
OA and the nine-point circle. Define
B″ and
C″ cyclically. The lines
AA″ ,
BB″ ,
CC″ concur in
X252 . (
Randy Hutson , June 7, 2019)
Search Value (6, 9, 13): \(2.767922111553\)
▸ Alternate Coordinates1 form
Trilinears: \(h(A,B,C) : h(B,C,A) : h(C,A,B)\) , where \(h(A,B,C) = \cos A \sec(3A) \sec(B - C)\) (Manol Iliev, 4/01/07)
▸ Incident Central Lines3 lines
{3 930 54 140 93 186
▸ Functional Relations3 relations
Notes
Let HA be the reflection of X4 BC X4 HB and HC cyclically. The lines AHA , BHB , CHC concur in X252
