X67
ISOGONAL CONJUGATE OF X23
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) { 69, 3448 }: Central Line L(69, 3448) { 6, 125 }: Central Line L(6, 125) { 110, 141 }: Central Line L(110, 141) { 50, 248 }: Central Line L(50, 248) { 4, 338 }: Central Line L(4, 338)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_3448 = ISOGONAL CONJUGATE OF X(3447) (1.1090, -0.4958) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_125 = CENTER OF JERABEK HYPERBOLA (0.6803, 0.0260) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_141 = COMPLEMENT OF SYMMEDIAN POINT (0.5061, 0.1663) X_50 = -CEVA CONJUGATE OF X(184) (2.1559, 2.8368) X_248 = -LINE CONJUGATE OF X(132) (0.8229, -0.0790) X_4 = ORTHOCENTER (0.6122, 0.3958) X_338 = CEVAPOINT OF X(115) AND X(125) (0.6941, 0.0587) X_67 = ISOGONAL CONJUGATE OF X(23) (0.7606, -0.2152)
A
B
C
X69
X3448
X6
X125
X110
X141
X50
X248
X4
X338
X67
(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) { 69, 3448 }: Central Line L(69, 3448) { 6, 125 }: Van Aubel Line { 110, 141 }: Central Line L(110, 141) { 50, 248 }: Central Line L(50, 248) { 4, 338 }: Van Aubel Line
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3600, 0.0000) X_3448 = ISOGONAL CONJUGATE OF X(3447) (0.9200, -0.4800) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_125 = CENTER OF JERABEK HYPERBOLA (0.6400, 0.0000) X_110 = FOCUS OF KIEPERT PARABOLA (0.3600, 0.4800) X_141 = COMPLEMENT OF SYMMEDIAN POINT (0.5000, 0.1200) X_50 = -CEVA CONJUGATE OF X(184) (2.2857, 3.0612) X_248 = -LINE CONJUGATE OF X(132) (0.7596, 0.0000) X_4 = ORTHOCENTER (0.6400, 0.4800) X_338 = CEVAPOINT OF X(115) AND X(125) (0.6400, 0.0349) X_67 = ISOGONAL CONJUGATE OF X(23) (0.6400, -0.2400)
A
B
C
X69
X3448
X6
X125
X110
X141
X50
X248
X4
X338
X67
(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) { 69, 3448 }: Central Line L(69, 3448) { 6, 125 }: Van Aubel Line { 110, 141 }: Central Line L(110, 141) { 50, 248 }: Central Line L(50, 248) { 4, 338 }: Van Aubel Line
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3820, 0.0000) X_3448 = ISOGONAL CONJUGATE OF X(3447) (0.8541, -0.4859) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_125 = CENTER OF JERABEK HYPERBOLA (0.6180, 0.0000) X_110 = FOCUS OF KIEPERT PARABOLA (0.3820, 0.4859) X_141 = COMPLEMENT OF SYMMEDIAN POINT (0.5000, 0.1215) X_50 = -CEVA CONJUGATE OF X(184) (2.6180, 4.3593) X_248 = -LINE CONJUGATE OF X(132) (0.7236, 0.0000) X_4 = ORTHOCENTER (0.6180, 0.4859) X_338 = CEVAPOINT OF X(115) AND X(125) (0.6180, 0.0256) X_67 = ISOGONAL CONJUGATE OF X(23) (0.6180, -0.2429)
A
B
C
X69
X3448
X6
X125
X110
X141
X50
X248
X4
X338
X67
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) { 69, 3448 }: Central Line L(69, 3448) { 6, 125 }: Central Line L(6, 125) { 110, 141 }: Central Line L(110, 141) { 50, 248 }: Central Line L(50, 248) { 4, 338 }: Central Line L(4, 338)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.3185, -0.0509) X_3448 = ISOGONAL CONJUGATE OF X(3447) (0.6382, -0.3115) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_125 = CENTER OF JERABEK HYPERBOLA (0.5678, -0.0079) X_110 = FOCUS OF KIEPERT PARABOLA (0.4975, 0.2958) X_141 = COMPLEMENT OF SYMMEDIAN POINT (0.4879, 0.0573) X_50 = -CEVA CONJUGATE OF X(184) (0.5919, -0.0176) X_248 = -LINE CONJUGATE OF X(132) (0.6841, 0.1154) X_4 = ORTHOCENTER (0.6331, 0.8294) X_338 = CEVAPOINT OF X(115) AND X(125) (0.5073, 0.0075) X_67 = ISOGONAL CONJUGATE OF X(23) (0.4783, -0.1812)
A
B
C
X69
X3448
X6
X125
X110
X141
X50
X248
X4
X338
X67
Kimberling (6, 9, 13) | AB = 1
X67 (ISOGONAL CONJUGATE OF X23 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(bc/(b^{4} + c^{4} - a^{4} - b^{2}c^{2}) : :\)
Barycentrics: \(1/(b^{4} + c^{4} - a^{4} - b^{2}c^{2}) : :\)
Search Value (6, 9, 13): \(5.987634908299\)
▸ Incident Central Lines10 lines
{3 542 4 338 6 125 50 248 74 935 110 141 265 511 290 340 524 858 526 879
▸ Functional Relations15 relations
Notes
Let A' be the reflection in BC of the A -vertex of the antipedal triangle of X6 B' and C' cyclically. The circumcircles of A'BC , B'CA , C'AB concur at X67 A' be the reflection of X6 BC , and define B' and C' cyclically. The circumcircles of A'BC , B'CA , C'AB concur in X67 Randy Hutson , November 18, 2015)
