X953
ISOGONAL CONJUGATE OF X952
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) { 901, 3 }: Central Line L(901, 3) { 36, 109 }: Central Line L(36, 109) { 100, 517 }: Central Line L(100, 517) { 104, 513 }: Central Line L(104, 513) { 110, 859 }: Central Line L(110, 859)
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_3 = CIRCUMCENTER (0.5000, 0.1021) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.3333, 1.6330) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.3636, 0.5938) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_517 = ISOGONAL CONJUGATE OF X(104) (0.8180, 0.7154) X_104 = ANTIPODE OF X(100) (0.8571, -0.2624) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0529, -0.0806) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_859 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(81) (1.1279, 1.7448) X_953 = ISOGONAL CONJUGATE OF X(952) (0.0000, 0.2041)
A
B
C
X901
X3
X36
X109
X100
X517
X104
X513
X110
X859
X953
(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) { 901, 3 }: Central Line L(901, 3) { 36, 109 }: Central Line L(36, 109) { 100, 517 }: Central Line L(100, 517) { 104, 513 }: Central Line L(104, 513) { 110, 859 }: Central Line L(110, 859)
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_3 = CIRCUMCENTER (0.5000, 0.0000) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.0000, 1.0000) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.5000, 0.5000) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2000, 0.4000) X_517 = ISOGONAL CONJUGATE OF X(104) (0.8038, 0.6743) X_104 = ANTIPODE OF X(100) (0.8000, -0.4000) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0610, -0.0971) X_110 = FOCUS OF KIEPERT PARABOLA (0.3600, 0.4800) X_859 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(81) (0.8182, 1.0909) X_953 = ISOGONAL CONJUGATE OF X(952) (0.0000, 0.0000)
A
B
C
X901
X3
X36
X109
X100
X517
X104
X513
X110
X859
X953
(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) { 901, 3 }: Central Line L(901, 3) { 36, 109 }: Central Line L(36, 109) { 100, 517 }: Central Line L(100, 517) { 104, 513 }: Central Line L(104, 513) { 110, 859 }: Central Line L(110, 859)
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_3 = CIRCUMCENTER (0.5000, 0.0000) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (0.9387, 1.0546) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.5000, 0.5000) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.2386, 0.4263) X_517 = ISOGONAL CONJUGATE OF X(104) (0.7602, 0.6929) X_104 = ANTIPODE OF X(100) (0.7614, -0.4263) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0703, -0.0589) X_110 = FOCUS OF KIEPERT PARABOLA (0.3820, 0.4859) X_859 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(81) (0.7729, 1.1235) X_953 = ISOGONAL CONJUGATE OF X(952) (0.0134, -0.1152)
A
B
C
X901
X3
X36
X109
X100
X517
X104
X513
X110
X859
X953
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) { 901, 3 }: Central Line L(901, 3) { 36, 109 }: Central Line L(36, 109) { 100, 517 }: Central Line L(100, 517) { 104, 513 }: Central Line L(104, 513) { 110, 859 }: Central Line L(110, 859)
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_3 = CIRCUMCENTER (0.5000, -0.2747) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (0.7120, 0.4690) X_109 = Ψ(INCENTER, CIRCUMCENTER) (0.6048, 0.2861) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.3206, 0.2669) X_517 = ISOGONAL CONJUGATE OF X(104) (0.7243, 0.7243) X_104 = ANTIPODE OF X(100) (0.6794, -0.8162) X_513 = ISOGONAL CONJUGATE OF X(100) (1.1753, -0.0865) X_110 = FOCUS OF KIEPERT PARABOLA (0.4975, 0.2958) X_859 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(81) (0.5937, 0.5028) X_953 = ISOGONAL CONJUGATE OF X(952) (0.0620, -0.6402)
A
B
C
X901
X3
X36
X109
X100
X517
X104
X513
X110
X859
X953
Kimberling (6, 9, 13) | AB = 1
X953 (ISOGONAL CONJUGATE OF X952 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(a/[2a^{4} - 2a^{3}(b + c) - a^{2}(b^{2} - 4bc + c^{2}) + (2a - b - c)(b - c)(b^{2} - c^{2})] : :\)
Search Value (6, 9, 13): \(14.014257686059\)
X953
▸ Incident Central Lines5 lines
{3 901 36 109 100 517 104 513 110 859
▸ Functional Relations12 relations
Notes
Let L be the line tangent to the incircle and the nine-point circle; i.e., the line X11 X244 LA be the reflection of L in BC , and define LB and LC cyclically. Let A″ = LB ∩LC , and define B″ and C″ cyclically. The lines AA″ , BB″ , CC″ concur in X953 Randy Hutson , 9/23/2011)
