X1268
ISOTOMIC CONJUGATE OF X1125
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) { 7, 12 }: Central Line L(7, 12) { 10, 86 }: Central Line L(10, 86) { 2, 594 }: Central Line L(2, 594) { 310, 313 }: Central Line L(310, 313) { 75, 1089 }: Central Line L(75, 1089) { 190, 1213 }: Central Line L(190, 1213)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1125 = COMPLEMENT OF X(10) (0.5459, 0.2083) X_7 = GERGONNE POINT (0.5903, 0.2769) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_86 = CEVAPOINT OF INCENTER AND CENTROID (0.5534, 0.2171) X_2 = CENTROID (0.5374, 0.2000) X_594 = ISOGONAL CONJUGATE OF X(593) (0.5025, 0.1672) X_310 = ISOTOMIC CONJUGATE OF X(42) (0.4762, 0.1536) X_313 = ISOTOMIC CONJUGATE OF X(58) (0.4368, 0.1249) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4988, 0.1682) X_1089 = TRILINEAR SQUARE OF X(10) (0.4606, 0.1381) X_190 = YFF PARABOLIC POINT (0.0748, 0.3999) X_1213 = ISOGONAL CONJUGATE OF X(1171) (0.5294, 0.1914) X_1268 = ISOTOMIC CONJUGATE OF X(1125) (0.5286, 0.1917)
A
B
C
X1125
X7
X12
X10
X86
X2
X594
X310
X313
X75
X1089
X190
X1213
X1268
(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) { 7, 12 }: Central Line L(7, 12) { 10, 86 }: Central Line L(10, 86) { 2, 594 }: Central Line L(2, 594) { 310, 313 }: Central Line L(310, 313) { 75, 1089 }: Central Line L(75, 1089) { 190, 1213 }: Central Line L(190, 1213)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_1125 = COMPLEMENT OF X(10) (0.5600, 0.1700) X_7 = GERGONNE POINT (0.6218, 0.2618) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5867, 0.2178) X_10 = SPIEKER CENTER (0.5200, 0.1400) X_86 = CEVAPOINT OF INCENTER AND CENTROID (0.5711, 0.1809) X_2 = CENTROID (0.5467, 0.1600) X_594 = ISOGONAL CONJUGATE OF X(593) (0.4915, 0.1212) X_310 = ISOTOMIC CONJUGATE OF X(42) (0.4433, 0.1060) X_313 = ISOTOMIC CONJUGATE OF X(58) (0.3816, 0.0755) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4826, 0.1226) X_1089 = TRILINEAR SQUARE OF X(10) (0.4218, 0.0891) X_190 = YFF PARABOLIC POINT (0.0933, 0.3200) X_1213 = ISOGONAL CONJUGATE OF X(1171) (0.5345, 0.1495) X_1268 = ISOTOMIC CONJUGATE OF X(1125) (0.5327, 0.1502)
A
B
C
X1125
X7
X12
X10
X86
X2
X594
X310
X313
X75
X1089
X190
X1213
X1268
(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) { 7, 12 }: Central Line L(7, 12) { 10, 86 }: Central Line L(10, 86) { 2, 594 }: Central Line L(2, 594) { 310, 313 }: Central Line L(310, 313) { 75, 1089 }: Central Line L(75, 1089) { 190, 1213 }: Central Line L(190, 1213)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_1125 = COMPLEMENT OF X(10) (0.5505, 0.1720) X_7 = GERGONNE POINT (0.6026, 0.2652) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5729, 0.2203) X_10 = SPIEKER CENTER (0.5170, 0.1419) X_86 = CEVAPOINT OF INCENTER AND CENTROID (0.5599, 0.1831) X_2 = CENTROID (0.5393, 0.1620) X_594 = ISOGONAL CONJUGATE OF X(593) (0.4931, 0.1231) X_310 = ISOTOMIC CONJUGATE OF X(42) (0.4532, 0.1089) X_313 = ISOTOMIC CONJUGATE OF X(58) (0.4007, 0.0784) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4859, 0.1249) X_1089 = TRILINEAR SQUARE OF X(10) (0.4347, 0.0916) X_190 = YFF PARABOLIC POINT (0.1322, 0.3610) X_1213 = ISOGONAL CONJUGATE OF X(1171) (0.5291, 0.1514) X_1268 = ISOTOMIC CONJUGATE OF X(1125) (0.5277, 0.1521)
A
B
C
X1125
X7
X12
X10
X86
X2
X594
X310
X313
X75
X1089
X190
X1213
X1268
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) { 7, 12 }: Central Line L(7, 12) { 10, 86 }: Central Line L(10, 86) { 2, 594 }: Central Line L(2, 594) { 310, 313 }: Central Line L(310, 313) { 75, 1089 }: Central Line L(75, 1089) { 190, 1213 }: Central Line L(190, 1213)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_1125 = COMPLEMENT OF X(10) (0.5621, 0.1025) X_7 = GERGONNE POINT (0.6288, 0.2114) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.6001, 0.1762) X_10 = SPIEKER CENTER (0.5089, 0.0750) X_86 = CEVAPOINT OF INCENTER AND CENTROID (0.5757, 0.1133) X_2 = CENTROID (0.5444, 0.0934) X_594 = ISOGONAL CONJUGATE OF X(593) (0.4705, 0.0589) X_310 = ISOTOMIC CONJUGATE OF X(42) (0.3899, 0.0479) X_313 = ISOTOMIC CONJUGATE OF X(58) (0.3112, 0.0266) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4506, 0.0607) X_1089 = TRILINEAR SQUARE OF X(10) (0.3698, 0.0351) X_190 = YFF PARABOLIC POINT (0.1548, 0.2119) X_1213 = ISOGONAL CONJUGATE OF X(1171) (0.5287, 0.0834) X_1268 = ISOTOMIC CONJUGATE OF X(1125) (0.5254, 0.0845)
A
B
C
X1125
X7
X12
X10
X86
X2
X594
X310
X313
X75
X1089
X190
X1213
X1268
Kimberling (6, 9, 13) | AB = 1
X1268 (ISOTOMIC CONJUGATE OF X1125 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(1/(a^{2}f(a,b,c)) : 1/(b^{2}f(b,c,a)) : 1/(c^{2}f(c,a,b))\) , where f(a,b,c) is as in
X1125 )
Search Value (6, 9, 13): \(2.870294405041\)
▸ Incident Central Lines7 lines
{2 594 7 12 10 86 75 1089 190 1213 310 313 333 1171
▸ Functional Relations6 relations
Notes
Let A22 B22 C22 be Gemini triangle 22. Let A' be the perspector of conic {{A , B , C ,B22 ,C22 }}, and define B' and C' cyclically. The lines AA' , BB' , CC' concur in X1268 Randy Hutson , January 15, 2019)
