X3417
X1 -VERTEX CONJUGATE OF X4
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) { 24, 56 }: Central Line L(24, 56) { 36, 47 }: Central Line L(36, 47) { 48, 573 }: Central Line L(48, 573) { 4, 2217 }: Central Line L(4, 2217)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_355 = FUHRMANN CENTER (0.5408, 0.2645) X_24 = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE (0.6627, 0.5278) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6316, 0.3438) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.3333, 1.6330) X_47 = -BETH CONJUGATE OF X(34) (0.7279, 0.4389) X_48 = CROSSPOINT OF X(1) AND X(63) (0.5412, 0.1246) X_573 = ORTHOHARMONIC OF X(58) (0.4486, 0.0172) X_4 = ORTHOCENTER (0.6122, 0.3958) X_2217 = -ISOCONJUGATE OF X(573) (0.6164, 0.5653) X_3417 = -VERTEX CONJUGATE OF X(4) (0.6075, 0.2015)
A
B
C
X355
X24
X56
X36
X47
X48
X573
X4
X2217
X3417
(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) { 24, 56 }: Central Line L(24, 56) { 36, 47 }: Central Line L(36, 47) { 48, 573 }: Central Line L(48, 573) { 4, 2217 }: Central Line L(4, 2217)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_355 = FUHRMANN CENTER (0.5400, 0.2800) X_24 = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE (0.6400, 0.4800) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6667, 0.3333) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.0000, 1.0000) X_47 = -BETH CONJUGATE OF X(34) (0.8182, 0.4545) X_48 = CROSSPOINT OF X(1) AND X(63) (0.5714, 0.0000) X_573 = ORTHOHARMONIC OF X(58) (0.4255, -0.1277) X_4 = ORTHOCENTER (0.6400, 0.4800) X_2217 = -ISOCONJUGATE OF X(573) (0.6000, 0.7000) X_3417 = -VERTEX CONJUGATE OF X(4) (0.6667, 0.3333)
A
B
C
X355
X24
X56
X36
X47
X48
X573
X4
X2217
X3417
(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) { 24, 56 }: Central Line L(24, 56) { 36, 47 }: Central Line L(36, 47) { 48, 573 }: Central Line L(48, 573) { 4, 2217 }: Central Line L(4, 2217)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_355 = FUHRMANN CENTER (0.5340, 0.2838) X_24 = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE (0.6180, 0.4859) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6411, 0.3392) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (0.9387, 1.0546) X_47 = -BETH CONJUGATE OF X(34) (0.7729, 0.4673) X_48 = CROSSPOINT OF X(1) AND X(63) (0.5599, 0.0000) X_573 = ORTHOHARMONIC OF X(58) (0.4375, -0.1285) X_4 = ORTHOCENTER (0.6180, 0.4859) X_2217 = -ISOCONJUGATE OF X(573) (0.5841, 0.7021) X_3417 = -VERTEX CONJUGATE OF X(4) (0.6411, 0.3392)
A
B
C
X355
X24
X56
X36
X47
X48
X573
X4
X2217
X3417
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) { 24, 56 }: Central Line L(24, 56) { 36, 47 }: Central Line L(36, 47) { 48, 573 }: Central Line L(48, 573) { 4, 2217 }: Central Line L(4, 2217)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_355 = FUHRMANN CENTER (0.5178, 0.4247) X_24 = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE (0.5555, 0.1854) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6494, 0.2495) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (0.7120, 0.4690) X_47 = -BETH CONJUGATE OF X(34) (3.6240, 3.4658) X_48 = CROSSPOINT OF X(1) AND X(63) (0.8073, -1.4298) X_573 = ORTHOHARMONIC OF X(58) (0.4096, -0.5275) X_4 = ORTHOCENTER (0.6331, 0.8294) X_2217 = -ISOCONJUGATE OF X(573) (0.5222, 0.6017) X_3417 = -VERTEX CONJUGATE OF X(4) (0.2019, -0.0560)
A
B
C
X355
X24
X56
X36
X47
X48
X573
X4
X2217
X3417
Kimberling (6, 9, 13) | AB = 1
X3417 (X1 -VERTEX CONJUGATE OF X4 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) , where
\(f(a,b,c) = a/[a^{2}vwyz - ux(bw + cv)(bz + cy)]\) , where
\(u : v : w\) =
X1 and
\(x : y : z\) =
X4 Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(6.874225075181\)
▸ Alternate Coordinates1 form
Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\) , where \(g(a,b,c) = a^{2}/[a^{4} - a^{3}(b + c) + 2a^{2}bc + a(b + c)(b - c)^{2} - (b^{2} c^{2})^{2}] (E\) . Danneels)
▸ Incident Central Lines4 lines
{4 2217 24 56 36 47 48 573
▸ Functional Relations1 relation
Notes
Let Ta Tb Tc be the intouch triangle . Let LA be the reflection of line Tb Tc in the perpedicular bisector of BC , and define LB and LC cyclically. Let A″ = LB ∩LC , and define B″ and C″ cyclically. The lines AA″ , BB″ , CC″ concur in X3417 hexyl triangle ’ is substituted for ‘intouch triangle’. (Randy Hutson , July 11, 2019)
