X900
CROSSDIFFERENCE OF X6 AND X101
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) { 11, 244 }: Central Line L(11, 244) { 100, 190 }: Central Line L(100, 190) { 30, 511 }: Central Line L(30, 511) { 37, 665 }: Central Line L(37, 665) { 335, 876 }: Central Line L(335, 876) { 673, 885 }: Central Line L(673, 885)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_901 = ISOGONAL CONJUGATE OF X(900) (1.0000, -0.0000) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_244 = -LINE CONJUGATE OF X(100) (0.7857, 0.1166) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_190 = YFF PARABOLIC POINT (0.0748, 0.3999) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2335, -0.3020) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5567, 0.2158) X_665 = CROSSDIFFERENCE OF X(2) AND X(11) (1.4579, 1.0988) X_335 = ISOTOMIC CONJUGATE OF X(239) (1.0387, 0.0318) X_876 = ISOTOMIC CONJUGATE OF X(874) (0.9873, -0.0185) X_673 = TRILINEAR POLE OF LINE X(1)X(514) (0.9471, -0.0769) X_885 = ISOTOMIC CONJUGATE OF X(883) (1.0518, 0.0256) X_900 = CROSSDIFFERENCE OF X(6) AND X(101) (0.9566, 0.6107)
A
B
C
X901
X11
X244
X100
X190
X30
X511
X37
X665
X335
X876
X673
X885
X900
(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) { 11, 244 }: Central Line L(11, 244) { 100, 190 }: Central Line L(100, 190) { 30, 511 }: Central Line L(30, 511) { 37, 665 }: Central Line L(37, 665) { 335, 876 }: Central Line L(335, 876) { 673, 885 }: Central Line L(673, 885)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_901 = ISOGONAL CONJUGATE OF X(900) (1.0000, -0.0000) X_11 = FEUERBACH POINT (0.7200, 0.0400) X_244 = -LINE CONJUGATE OF X(100) (0.8000, 0.1000) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2000, 0.4000) X_190 = YFF PARABOLIC POINT (0.0933, 0.3200) X_30 = EULER INFINITY POINT (0.7077, 0.7120) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2569, -0.3367) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5787, 0.1787) X_665 = CROSSDIFFERENCE OF X(2) AND X(11) (1.1915, 0.6383) X_335 = ISOTOMIC CONJUGATE OF X(239) (1.0641, 0.0374) X_876 = ISOTOMIC CONJUGATE OF X(874) (0.9830, -0.0234) X_673 = TRILINEAR POLE OF LINE X(1)X(514) (0.9248, -0.1018) X_885 = ISOTOMIC CONJUGATE OF X(883) (1.0945, 0.0255) X_900 = CROSSDIFFERENCE OF X(6) AND X(101) (1.0067, 0.5050)
A
B
C
X901
X11
X244
X100
X190
X30
X511
X37
X665
X335
X876
X673
X885
X900
(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) { 11, 244 }: Central Line L(11, 244) { 100, 190 }: Central Line L(100, 190) { 30, 511 }: Central Line L(30, 511) { 37, 665 }: Central Line L(37, 665) { 673, 885 }: Central Line L(673, 885) { 1635, 1644 }: Central Line L(1635, 1644)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_901 = ISOGONAL CONJUGATE OF X(900) (0.9866, 0.1152) X_11 = FEUERBACH POINT (0.6897, 0.0298) X_244 = -LINE CONJUGATE OF X(100) (0.7719, 0.0802) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2386, 0.4263) X_190 = YFF PARABOLIC POINT (0.1322, 0.3610) X_30 = EULER INFINITY POINT (0.6751, 0.7207) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2881, -0.3552) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5661, 0.1805) X_665 = CROSSDIFFERENCE OF X(2) AND X(11) (1.3090, 0.6360) X_673 = TRILINEAR POLE OF LINE X(1)X(514) (0.8833, -0.1403) X_885 = ISOTOMIC CONJUGATE OF X(883) (1.1639, 0.0318) X_1635 = TRIPOLAR CENTROID OF X(1) (1.8360, 1.1813) X_1644 = TRIPOLAR CENTROID OF X(190) (1.3279, 0.8698) X_900 = CROSSDIFFERENCE OF X(6) AND X(101) (1.0295, 0.4625)
A
B
C
X901
X11
X244
X100
X190
X30
X511
X37
X665
X673
X885
X1635
X1644
X900
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) { 11, 244 }: Central Line L(11, 244) { 100, 190 }: Central Line L(100, 190) { 30, 511 }: Central Line L(30, 511) { 37, 665 }: Central Line L(37, 665) { 335, 876 }: Central Line L(335, 876) { 673, 885 }: Central Line L(673, 885)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_901 = ISOGONAL CONJUGATE OF X(900) (0.9380, 0.0908) X_11 = FEUERBACH POINT (0.6563, 0.0066) X_244 = -LINE CONJUGATE OF X(100) (0.7876, 0.0501) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.3206, 0.2669) X_190 = YFF PARABOLIC POINT (0.1548, 0.2119) X_30 = EULER INFINITY POINT (0.6229, 0.7447) X_511 = ISOGONAL CONJUGATE OF X(98) (0.3235, -0.5244) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5913, 0.1097) X_665 = CROSSDIFFERENCE OF X(2) AND X(11) (1.0594, 0.2647) X_335 = ISOTOMIC CONJUGATE OF X(239) (1.0278, 0.0074) X_876 = ISOTOMIC CONJUGATE OF X(874) (0.9918, -0.0045) X_673 = TRILINEAR POLE OF LINE X(1)X(514) (0.8495, -0.1432) X_885 = ISOTOMIC CONJUGATE OF X(883) (1.3227, 0.0135) X_900 = CROSSDIFFERENCE OF X(6) AND X(101) (1.1671, 0.2997)
A
B
C
X901
X11
X244
X100
X190
X30
X511
X37
X665
X335
X876
X673
X885
X900
Kimberling (6, 9, 13) | AB = 1
X900 (CROSSDIFFERENCE OF X6 AND X101 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \((b - c)(b + c - 2a)/a : :\)
Barycentrics: \((b - c)(b + c - 2a) : (c - a)(c + a - 2b) : (a - b)(a + b - 2c)\)
▸ Incident Central Lines7 lines
{11 244 30 511 37 665 100 190 335 876 673 885 1635 1644
▸ Functional Relations16 relations
Notes
Let Ia be the reflection of X1 BC , and define Ib and Ic cyclically; then X900 X513 Ia Ib Ic . (Randy Hutson , February 10, 2016)
Let Ua be the line through X80 X80 Ub and Uc cyclically. Let Va be the reflection of BC in Ua , and define Vb and Vc cyclically. The lines Va , Vb , Vc are parallel, and they concur in X900 Angel Montesdeoca , June 30, 2017)
