X3062
ISOGONAL CONJUGATE OF X165
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) { 8, 144 }: Central Line L(8, 144) { 9, 165 }: Central Line L(9, 165) { 1, 971 }: Central Line L(1, 971) { 515, 1000 }: Central Line L(515, 1000) { 7, 1699 }: Central Line L(7, 1699) { 294, 1721 }: Central Line L(294, 1721)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4762, 0.0583) X_8 = NAGEL POINT (0.4694, 0.1333) X_144 = ANTICOMPLEMENT OF X(7) (0.4317, 0.0461) X_9 = MITTENPUNKT (0.5110, 0.1615) X_1 = INCENTER (0.5714, 0.2333) X_971 = INTERSECTION OF LINES X(3)X(9) AND X(4)X(7) (0.6441, 0.7770) X_515 = ISOGONAL CONJUGATE OF X(102) (0.6804, 0.7691) X_1000 = ISOGONAL CONJUGATE OF X(999) (0.5463, 0.1961) X_7 = GERGONNE POINT (0.5903, 0.2769) X_1699 = EXSIMILICENTER(BEVAN CIRCLE, NINE-POINT CIRCLE) (0.5986, 0.3416) X_294 = ISOGONAL CONJUGATE OF X(241) (0.9276, -0.0575) X_1721 = MIMOSA TRANSFORM OF X(55) (0.6494, 0.4136) X_3062 = ISOGONAL CONJUGATE OF X(165) (0.6154, 0.4711)
A
B
C
X165
X8
X144
X9
X1
X971
X515
X1000
X7
X1699
X294
X1721
X3062
(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) { 8, 144 }: Central Line L(8, 144) { 9, 165 }: Central Line L(9, 165) { 1, 971 }: Central Line L(1, 971) { 515, 1000 }: Central Line L(515, 1000) { 7, 1699 }: Central Line L(7, 1699) { 294, 1721 }: Central Line L(294, 1721)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4667, -0.0667) X_8 = NAGEL POINT (0.4400, 0.0800) X_144 = ANTICOMPLEMENT OF X(7) (0.3964, -0.0436) X_9 = MITTENPUNKT (0.5091, 0.1091) X_1 = INCENTER (0.6000, 0.2000) X_971 = INTERSECTION OF LINES X(3)X(9) AND X(4)X(7) (0.5944, 0.7330) X_515 = ISOGONAL CONJUGATE OF X(102) (0.6280, 0.7292) X_1000 = ISOGONAL CONJUGATE OF X(999) (0.5673, 0.1527) X_7 = GERGONNE POINT (0.6218, 0.2618) X_1699 = EXSIMILICENTER(BEVAN CIRCLE, NINE-POINT CIRCLE) (0.6267, 0.3867) X_294 = ISOGONAL CONJUGATE OF X(241) (0.8883, -0.0579) X_1721 = MIMOSA TRANSFORM OF X(55) (0.6960, 0.4720) X_3062 = ISOGONAL CONJUGATE OF X(165) (0.6364, 0.6364)
A
B
C
X165
X8
X144
X9
X1
X971
X515
X1000
X7
X1699
X294
X1721
X3062
(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) { 8, 144 }: Central Line L(8, 144) { 9, 165 }: Central Line L(9, 165) { 1, 971 }: Central Line L(1, 971) { 515, 1000 }: Central Line L(515, 1000) { 7, 1699 }: Central Line L(7, 1699) { 294, 1721 }: Central Line L(294, 1721)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4720, -0.0674) X_8 = NAGEL POINT (0.4499, 0.0817) X_144 = ANTICOMPLEMENT OF X(7) (0.4128, -0.0446) X_9 = MITTENPUNKT (0.5077, 0.1103) X_1 = INCENTER (0.5841, 0.2021) X_971 = INTERSECTION OF LINES X(3)X(9) AND X(4)X(7) (0.5795, 0.7356) X_515 = ISOGONAL CONJUGATE OF X(102) (0.6077, 0.7329) X_1000 = ISOGONAL CONJUGATE OF X(999) (0.5563, 0.1539) X_7 = GERGONNE POINT (0.6026, 0.2652) X_1699 = EXSIMILICENTER(BEVAN CIRCLE, NINE-POINT CIRCLE) (0.6067, 0.3913) X_294 = ISOGONAL CONJUGATE OF X(241) (0.8391, -0.0765) X_1721 = MIMOSA TRANSFORM OF X(55) (0.6657, 0.4802) X_3062 = ISOGONAL CONJUGATE OF X(165) (0.6149, 0.6434)
A
B
C
X165
X8
X144
X9
X1
X971
X515
X1000
X7
X1699
X294
X1721
X3062
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) { 8, 144 }: Central Line L(8, 144) { 9, 165 }: Central Line L(9, 165) { 1, 971 }: Central Line L(1, 971) { 515, 1000 }: Central Line L(515, 1000) { 7, 1699 }: Central Line L(7, 1699) { 294, 1721 }: Central Line L(294, 1721)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4615, -0.4096) X_8 = NAGEL POINT (0.4024, 0.0200) X_144 = ANTICOMPLEMENT OF X(7) (0.3756, -0.1427) X_9 = MITTENPUNKT (0.5022, 0.0343) X_1 = INCENTER (0.6154, 0.1300) X_971 = INTERSECTION OF LINES X(3)X(9) AND X(4)X(7) (0.5490, 0.7494) X_515 = ISOGONAL CONJUGATE OF X(102) (0.5610, 0.7492) X_1000 = ISOGONAL CONJUGATE OF X(999) (0.5915, 0.0805) X_7 = GERGONNE POINT (0.6288, 0.2114) X_1699 = EXSIMILICENTER(BEVAN CIRCLE, NINE-POINT CIRCLE) (0.6272, 0.5963) X_294 = ISOGONAL CONJUGATE OF X(241) (0.7790, -0.0299) X_1721 = MIMOSA TRANSFORM OF X(55) (0.7025, 0.6591) X_3062 = ISOGONAL CONJUGATE OF X(165) (0.6241, 1.3661)
A
B
C
X165
X8
X144
X9
X1
X971
X515
X1000
X7
X1699
X294
X1721
X3062
Kimberling (6, 9, 13) | AB = 1
X3062 (ISOGONAL CONJUGATE OF X165 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(1/[\tan(B/2) + \tan(C/2) - \tan(A/2)] :\)
Barycentrics: \(a/[\tan(B/2) + \tan(C/2) - \tan(A/2)] : :\)
Search Value (6, 9, 13): \(-11.151252270409\)
▸ Incident Central Lines12 lines
{1 971 7 1699 8 144 9 165 80 2093 269 2310 294 1721 515 1000 943 1490 1156 1445 1320 2801 2335 2947
▸ Functional Relations7 relations
Notes
In the plane of a triangle ABC , let
A' = A -excircle of ABFC, and define B' and C' cyclically; A1 = orthopole of BC wrt B'C' , and define B1 and C1 cyclically; Then X2 A1 B1 C1 -to-ABC orthology center, and X3062 ABC -to-A1 B1 C1 orthology center. (Ivan Pavlov, August 21, 2022)
