X313
ISOTOMIC CONJUGATE OF X58
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) { 10, 75 }: Central Line L(10, 75) { 71, 190 }: Central Line L(71, 190) { 42, 308 }: Central Line L(42, 308) { 92, 264 }: Central Line L(92, 264) { 12, 349 }: Central Line L(12, 349) { 80, 314 }: Central Line L(80, 314)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6107, 0.2850) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4988, 0.1682) X_71 = ISOGONAL CONJUGATE OF X(27) (0.4807, 0.0915) X_190 = YFF PARABOLIC POINT (0.0748, 0.3999) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.5880, 0.2495) X_308 = ISOTOMIC CONJUGATE OF X(39) (0.4915, 0.1700) X_92 = CEVAPOINT OF INCENTER AND CLAWSON POINT (0.5949, 0.3639) X_264 = ISOTOMIC CONJUGATE OF CIRCUMCENTER (0.5720, 0.3293) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_349 = ISOTOMIC CONJUGATE OF X(284) (0.5089, 0.1906) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.8980, -0.1000) X_314 = ISOTOMIC CONJUGATE OF X(65) (0.4457, 0.1206) X_313 = ISOTOMIC CONJUGATE OF X(58) (0.4368, 0.1249)
A
B
C
X58
X10
X75
X71
X190
X42
X308
X92
X264
X12
X349
X80
X314
X313
(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) { 10, 75 }: Central Line L(10, 75) { 71, 190 }: Central Line L(71, 190) { 42, 308 }: Central Line L(42, 308) { 12, 349 }: Central Line L(12, 349) { 80, 314 }: Central Line L(80, 314) { 321, 594 }: Central Line L(321, 594)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6522, 0.2609) X_10 = SPIEKER CENTER (0.5200, 0.1400) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4826, 0.1226) X_71 = ISOGONAL CONJUGATE OF X(27) (0.4706, 0.0000) X_190 = YFF PARABOLIC POINT (0.0933, 0.3200) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.6250, 0.2188) X_308 = ISOTOMIC CONJUGATE OF X(39) (0.4655, 0.1249) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5867, 0.2178) X_349 = ISOTOMIC CONJUGATE OF X(284) (0.4971, 0.1557) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.8400, -0.1200) X_314 = ISOTOMIC CONJUGATE OF X(65) (0.3996, 0.0678) X_321 = ISOTOMIC CONJUGATE OF X(81) (0.4525, 0.1050) X_594 = ISOGONAL CONJUGATE OF X(593) (0.4915, 0.1212) X_313 = ISOTOMIC CONJUGATE OF X(58) (0.3816, 0.0755)
A
B
C
X58
X10
X75
X71
X190
X42
X308
X12
X349
X80
X314
X321
X594
X313
(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) { 10, 75 }: Central Line L(10, 75) { 71, 190 }: Central Line L(71, 190) { 42, 308 }: Central Line L(42, 308) { 12, 349 }: Central Line L(12, 349) { 80, 314 }: Central Line L(80, 314) { 321, 594 }: Central Line L(321, 594)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6285, 0.2645) X_10 = SPIEKER CENTER (0.5170, 0.1419) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4859, 0.1249) X_71 = ISOGONAL CONJUGATE OF X(27) (0.4753, 0.0000) X_190 = YFF PARABOLIC POINT (0.1322, 0.3610) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.6052, 0.2210) X_308 = ISOTOMIC CONJUGATE OF X(39) (0.4721, 0.1282) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5729, 0.2203) X_349 = ISOTOMIC CONJUGATE OF X(284) (0.4983, 0.1589) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.7953, -0.1425) X_314 = ISOTOMIC CONJUGATE OF X(65) (0.4159, 0.0699) X_321 = ISOTOMIC CONJUGATE OF X(81) (0.4606, 0.1074) X_594 = ISOGONAL CONJUGATE OF X(593) (0.4931, 0.1231) X_313 = ISOTOMIC CONJUGATE OF X(58) (0.4007, 0.0784)
A
B
C
X58
X10
X75
X71
X190
X42
X308
X12
X349
X80
X314
X321
X594
X313
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) { 10, 75 }: Central Line L(10, 75) { 71, 190 }: Central Line L(71, 190) { 42, 308 }: Central Line L(42, 308) { 92, 264 }: Central Line L(92, 264) { 12, 349 }: Central Line L(12, 349) { 80, 314 }: Central Line L(80, 314)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6639, 0.1838) X_10 = SPIEKER CENTER (0.5089, 0.0750) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4506, 0.0607) X_71 = ISOGONAL CONJUGATE OF X(27) (0.4699, -0.1616) X_190 = YFF PARABOLIC POINT (0.1548, 0.2119) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.6461, 0.1459) X_308 = ISOTOMIC CONJUGATE OF X(39) (0.4157, 0.0638) X_92 = CEVAPOINT OF INCENTER AND CLAWSON POINT (-0.2072, -2.1182) X_264 = ISOTOMIC CONJUGATE OF CIRCUMCENTER (0.2381, -0.2757) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.6001, 0.1762) X_349 = ISOTOMIC CONJUGATE OF X(284) (0.4862, 0.1172) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.6972, -0.1168) X_314 = ISOTOMIC CONJUGATE OF X(65) (0.3421, 0.0151) X_313 = ISOTOMIC CONJUGATE OF X(58) (0.3112, 0.0266)
A
B
C
X58
X10
X75
X71
X190
X42
X308
X92
X264
X12
X349
X80
X314
X313
Kimberling (6, 9, 13) | AB = 1
X313 (ISOTOMIC CONJUGATE OF X58 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \((b + c)b^{3}c^{3} : (c + a)c^{3}a^{3} : (a + b)a^{3}b^{3}\)
Barycentrics: \((b + c)b^{2}c^{2} : (c + a)c^{2}a^{2} : (a + b)a^{2}b^{2}\)
Search Value (6, 9, 13): \(5.158734620701\)
▸ Alternate Coordinates1 form
▸ Incident Central Lines8 lines
{10 75 12 349 42 308 71 190 80 314 92 264 321 594 561 696
▸ Functional Relations9 relations
Notes
Let A40 B40 C40 be Gemini triangle 40. Let A' be the center of conic {A , B , C ,B40 ,C40 }}, and define B' and C' cyclically. The lines AA' , BB' , CC' concur in X313 Randy Hutson , January 15, 2019)
