X334
ISOTOMIC CONJUGATE OF X238
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) { 12, 85 }: Central Line L(12, 85) { 75, 141 }: Central Line L(75, 141) { 10, 274 }: Central Line L(10, 274) { 2, 292 }: Central Line L(2, 292) { 150, 295 }: Central Line L(150, 295) { 320, 660 }: Central Line L(320, 660)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_238 = -LINE CONJUGATE OF X(37) (1.6190, 1.4775) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_85 = ISOTOMIC CONJUGATE OF X(9) (0.5610, 0.2421) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4988, 0.1682) X_141 = COMPLEMENT OF SYMMEDIAN POINT (0.5061, 0.1663) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_274 = ISOGONAL CONJUGATE OF X(213) (0.5168, 0.1843) X_2 = CENTROID (0.5374, 0.2000) X_292 = -HIRST INVERSE OF X(291) (1.0127, 0.0417) X_150 = REFLECTION OF X(20) IN X(103) (1.1837, -0.4499) X_295 = ISOGONAL CONJUGATE OF X(242) (1.0582, 0.0142) X_320 = ISOTOMIC CONJUGATE OF X(80) (-2.6735, -3.5993) X_660 = TRILINEAR POLE OF LINE X(1)X(39) (0.9483, -0.0755) X_334 = ISOTOMIC CONJUGATE OF X(238) (1.0545, 0.0278)
A
B
C
X238
X12
X85
X75
X141
X10
X274
X2
X292
X150
X295
X320
X660
X334
(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) { 12, 85 }: Central Line L(12, 85) { 75, 141 }: Central Line L(75, 141) { 10, 274 }: Central Line L(10, 274) { 2, 292 }: Central Line L(2, 292) { 150, 295 }: Central Line L(150, 295) { 320, 660 }: Central Line L(320, 660)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_238 = -LINE CONJUGATE OF X(37) (1.2667, 0.8667) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5867, 0.2178) X_85 = ISOTOMIC CONJUGATE OF X(9) (0.5801, 0.2201) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4826, 0.1226) X_141 = COMPLEMENT OF SYMMEDIAN POINT (0.5000, 0.1200) X_10 = SPIEKER CENTER (0.5200, 0.1400) X_274 = ISOGONAL CONJUGATE OF X(213) (0.5114, 0.1416) X_2 = CENTROID (0.5467, 0.1600) X_292 = -HIRST INVERSE OF X(291) (1.0074, 0.0540) X_150 = REFLECTION OF X(20) IN X(103) (1.0246, -0.4431) X_295 = ISOGONAL CONJUGATE OF X(242) (1.1000, 0.0000) X_320 = ISOTOMIC CONJUGATE OF X(80) (-1.5600, -1.9200) X_660 = TRILINEAR POLE OF LINE X(1)X(39) (0.9289, -0.0978) X_334 = ISOTOMIC CONJUGATE OF X(238) (1.1053, 0.0314)
A
B
C
X238
X12
X85
X75
X141
X10
X274
X2
X292
X150
X295
X320
X660
X334
(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) { 12, 85 }: Central Line L(12, 85) { 75, 141 }: Central Line L(75, 141) { 10, 274 }: Central Line L(10, 274) { 2, 292 }: Central Line L(2, 292) { 150, 295 }: Central Line L(150, 295) { 320, 660 }: Central Line L(320, 660)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_238 = -LINE CONJUGATE OF X(37) (1.2021, 0.9450) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5729, 0.2203) X_85 = ISOTOMIC CONJUGATE OF X(9) (0.5676, 0.2231) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4859, 0.1249) X_141 = COMPLEMENT OF SYMMEDIAN POINT (0.5000, 0.1215) X_10 = SPIEKER CENTER (0.5170, 0.1419) X_274 = ISOGONAL CONJUGATE OF X(213) (0.5101, 0.1439) X_2 = CENTROID (0.5393, 0.1620) X_292 = -HIRST INVERSE OF X(291) (1.0000, 0.0000) X_150 = REFLECTION OF X(20) IN X(103) (0.9450, -0.4592) X_295 = ISOGONAL CONJUGATE OF X(242) (1.0000, 0.0000) X_320 = ISOTOMIC CONJUGATE OF X(80) (-1.4042, -2.1471) X_660 = TRILINEAR POLE OF LINE X(1)X(39) (1.0000, 0.0000) X_334 = ISOTOMIC CONJUGATE OF X(238) (1.0000, 0.0000)
A
B
C
X238
X12
X85
X75
X141
X10
X274
X2
X292
X150
X295
X320
X660
X334
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) { 12, 85 }: Central Line L(12, 85) { 75, 141 }: Central Line L(75, 141) { 10, 274 }: Central Line L(10, 274) { 2, 292 }: Central Line L(2, 292) { 150, 295 }: Central Line L(150, 295) { 320, 660 }: Central Line L(320, 660)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_238 = -LINE CONJUGATE OF X(37) (0.9397, 0.4041) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.6001, 0.1762) X_85 = ISOTOMIC CONJUGATE OF X(9) (0.5911, 0.1796) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4506, 0.0607) X_141 = COMPLEMENT OF SYMMEDIAN POINT (0.4879, 0.0573) X_10 = SPIEKER CENTER (0.5089, 0.0750) X_274 = ISOGONAL CONJUGATE OF X(213) (0.4902, 0.0774) X_2 = CENTROID (0.5444, 0.0934) X_292 = -HIRST INVERSE OF X(291) (0.9966, 0.0147) X_150 = REFLECTION OF X(20) IN X(103) (0.7368, -0.3068) X_295 = ISOGONAL CONJUGATE OF X(242) (1.0388, -0.0067) X_320 = ISOTOMIC CONJUGATE OF X(80) (-0.5706, -0.8023) X_660 = TRILINEAR POLE OF LINE X(1)X(39) (0.9885, -0.0258) X_334 = ISOTOMIC CONJUGATE OF X(238) (1.0508, 0.0053)
A
B
C
X238
X12
X85
X75
X141
X10
X274
X2
X292
X150
X295
X320
X660
X334
Kimberling (6, 9, 13) | AB = 1
X334 (ISOTOMIC CONJUGATE OF X238 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(b^{2}c^{2}/(a^{2} - bc) : c^{2}a^{2}/(b^{2} - ca) : a^{2}b^{2}/(c^{2} - ab)\)
Barycentrics: bc/(
\(a^{2}\) - bc) : ca/(
\(b^{2}\) - ca) : ab/(
\(c^{2}\) - ab) Let
A31 B31 C31 be Gemini triangle 31. Let
A' be the perspector of conic {
A ,
B ,
C ,
B31 ,
C31 }}, and define
B' and
C' cyclically. The lines
AA' ,
BB' ,
CC' concur in
X334 . (
Randy Hutson , January 15, 2019)
Search Value (6, 9, 13): \(-0.455299758091\)
▸ Incident Central Lines9 lines
{2 292 10 274 12 85 75 141 76 1089 150 295 320 660 741 839 767 813
▸ Functional Relations5 relations
