X893
X238 -CROSS CONJUGATE OF X292
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) { 9, 43 }: Central Line L(9, 43) { 19, 232 }: Central Line L(19, 232) { 171, 292 }: Central Line L(171, 292) { 239, 257 }: Central Line L(239, 257) { 42, 694 }: Central Line L(42, 694) { 100, 733 }: Central Line L(100, 733)
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_292 = -HIRST INVERSE OF X(291) (1.0127, 0.0417) X_9 = MITTENPUNKT (0.5110, 0.1615) X_43 = -CEVA CONJUGATE OF X(1) (0.6358, 0.2963) X_19 = CLAWSON POINT (0.6246, 0.4244) X_232 = -CEVA CONJUGATE OF X(132) (0.6919, 0.7779) X_171 = {X(2), X(31)}-HARMONIC CONJUGATE OF X(238) (0.5819, 0.2525) X_239 = -LINE CONJUGATE OF X(42) (4.2109, 3.7992) X_257 = ISOGONAL CONJUGATE OF X(172) (0.5240, 0.1822) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.5880, 0.2495) X_694 = ISOGONAL CONJUGATE OF X(385) (1.0179, 0.0378) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_733 = EVEN (- 2, 2) CIRCUMCIRCLE POINT (1.0071, 0.0450) X_893 = -CROSS CONJUGATE OF X(292) (0.5910, 0.2480)
A
B
C
X238
X292
X9
X43
X19
X232
X171
X239
X257
X42
X694
X100
X733
X893
(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) { 9, 43 }: Central Line L(9, 43) { 171, 292 }: Central Line L(171, 292) { 239, 257 }: Central Line L(239, 257) { 42, 694 }: Central Line L(42, 694) { 100, 733 }: Central Line L(100, 733) { 55, 869 }: Central Line L(55, 869)
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_292 = -HIRST INVERSE OF X(291) (1.0074, 0.0540) X_9 = MITTENPUNKT (0.5091, 0.1091) X_43 = -CEVA CONJUGATE OF X(1) (0.6941, 0.2706) X_171 = {X(2), X(31)}-HARMONIC CONJUGATE OF X(238) (0.6121, 0.2242) X_239 = -LINE CONJUGATE OF X(42) (3.1067, 2.0800) X_257 = ISOGONAL CONJUGATE OF X(172) (0.5285, 0.1384) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.6250, 0.2188) X_694 = ISOGONAL CONJUGATE OF X(385) (1.0176, 0.0462) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2000, 0.4000) X_733 = EVEN (- 2, 2) CIRCUMCIRCLE POINT (0.9964, 0.0602) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5714, 0.1429) X_869 = INTERSECTION OF LINES X(1)X(2) AND X(31)X(32) (0.6480, 0.2360) X_893 = -CROSS CONJUGATE OF X(292) (0.6315, 0.2159)
A
B
C
X238
X292
X9
X43
X171
X239
X257
X42
X694
X100
X733
X55
X869
X893
(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) { 9, 43 }: Central Line L(9, 43) { 171, 292 }: Central Line L(171, 292) { 239, 257 }: Central Line L(239, 257) { 42, 694 }: Central Line L(42, 694) { 100, 733 }: Central Line L(100, 733) { 55, 869 }: Central Line L(55, 869)
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_292 = -HIRST INVERSE OF X(291) (1.0000, 0.0000) X_9 = MITTENPUNKT (0.5077, 0.1103) X_43 = -CEVA CONJUGATE OF X(1) (0.6641, 0.2739) X_171 = {X(2), X(31)}-HARMONIC CONJUGATE OF X(238) (0.5944, 0.2271) X_239 = -LINE CONJUGATE OF X(42) (2.8901, 2.2720) X_257 = ISOGONAL CONJUGATE OF X(172) (0.5239, 0.1399) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.6052, 0.2210) X_694 = ISOGONAL CONJUGATE OF X(385) (1.0000, 0.0000) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2386, 0.4263) X_733 = EVEN (- 2, 2) CIRCUMCIRCLE POINT (1.0000, 0.0000) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5599, 0.1439) X_869 = INTERSECTION OF LINES X(1)X(2) AND X(31)X(32) (0.6248, 0.2386) X_893 = -CROSS CONJUGATE OF X(292) (0.6106, 0.2180)
A
B
C
X238
X292
X9
X43
X171
X239
X257
X42
X694
X100
X733
X55
X869
X893
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) { 9, 43 }: Central Line L(9, 43) { 19, 232 }: Central Line L(19, 232) { 171, 292 }: Central Line L(171, 292) { 239, 257 }: Central Line L(239, 257) { 42, 694 }: Central Line L(42, 694) { 100, 733 }: Central Line L(100, 733)
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_292 = -HIRST INVERSE OF X(291) (0.9966, 0.0147) X_9 = MITTENPUNKT (0.5022, 0.0343) X_43 = -CEVA CONJUGATE OF X(1) (0.7237, 0.1860) X_19 = CLAWSON POINT (0.5737, 0.4687) X_232 = -CEVA CONJUGATE OF X(132) (0.6407, 0.2095) X_171 = {X(2), X(31)}-HARMONIC CONJUGATE OF X(238) (0.6224, 0.1547) X_239 = -LINE CONJUGATE OF X(42) (2.0489, 0.8704) X_257 = ISOGONAL CONJUGATE OF X(172) (0.5274, 0.0726) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.6461, 0.1459) X_694 = ISOGONAL CONJUGATE OF X(385) (1.0028, 0.0106) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.3206, 0.2669) X_733 = EVEN (- 2, 2) CIRCUMCIRCLE POINT (0.9896, 0.0181) X_893 = -CROSS CONJUGATE OF X(292) (0.6584, 0.1412)
A
B
C
X238
X292
X9
X43
X19
X232
X171
X239
X257
X42
X694
X100
X733
X893
Kimberling (6, 9, 13) | AB = 1
X893 (X238 -CROSS CONJUGATE OF X292 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(a/(a^{2} + bc) : :\)
Search Value (6, 9, 13): \(1.235229038235\)
▸ Incident Central Lines7 lines
{9 43 19 232 42 694 55 869 100 733 171 292 239 257
▸ Functional Relations12 relations
Notes
Let A34 B34 C34 be Gemini triangle 34. Let A' be the perspector of conic {A , B , C ,B34 ,C34 }}, and define B' and C' cyclically. The lines AA' , BB' , CC' concur in X893 Randy Hutson , January 15, 2019)
