[Kadir Altintas]:
Let ABC be a triangle, P a point and DEF the cevian triangle of P.
Denote:
Gamma_P = the circumconic through G_P, K_P = centroid, symmedian poin of DEF, resp.
1. For P = H, the Gamma_H passes through K = symmedian point of ABC
2. G_H = X(51), K_H = X(53)
Gamma_H passes through Xi for i =216,288,343.2052.2351
Questions
-- Is this conic known?
-- More points it passes through?
[Peter Moses]:
Hi Antreas,
a^2 (b^2-c^2) (a^2-b^2-c^2) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) y z + cyclic is a hyperbola that passes through
ABC, X{6,51,53,216,288,343,2052, 2351,11077,14582}.
------------------------------ ------------------------------ ---
Perspector: a^2 (b^2-c^2) (a^2-b^2-c^2) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4)::
------------------------------ ------------------------------ -
Center: a^2 (b-c)^2 (b+c)^2 (a^2-b^2-c^2) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^8-a^6 b^2-a^4 b^4+a^2 b^6-a^6 c^2+3 a^4 b^2 c^2-3 a^2 b^4 c^2+b^6 c^2-a^4 c^4-3 a^2 b^2 c^4-2 b^4 c^4+a^2 c^6+b^2 c^6)::
------------------------------ ------------------------------ -
Isogonal conjugate of line through X{2,95,97,233,275,317,577, 3087,4993,6709,8882,10311, 10313,10314,14590,14918}.
Best regards,
Peter Moses.
Denote:
Gamma_P = the circumconic through G_P, K_P = centroid, symmedian poin of DEF, resp.
1. For P = H, the Gamma_H passes through K = symmedian point of ABC
2. G_H = X(51), K_H = X(53)
Gamma_H passes through Xi for i =216,288,343.2052.2351
Questions
-- Is this conic known?
-- More points it passes through?
[Peter Moses]:
Hi Antreas,
a^2 (b^2-c^2) (a^2-b^2-c^2) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) y z + cyclic is a hyperbola that passes through
ABC, X{6,51,53,216,288,343,2052, 2351,11077,14582}.
------------------------------ ------------------------------ ---
Perspector: a^2 (b^2-c^2) (a^2-b^2-c^2) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4)::
on lines {{4,15412},{130,3269},{137, 5099},{187,237},{403,523},{ 525,684},{826,3574},{878, 10547},{1157,1510},{1568,6368} ,{10254,14592},{14380,14483}}.
midpoint of X(4) and X(15412).
4 X[647] - X[9409].
isogonal of the isotomic of X(6368).
orthic isogonal conjugate of X(3269).
X(647)-daleth conjugate of X(3569).
X(i)-Ceva conjugate of X(j) for these (i,j): {{4, 3269}, {523, 12077}, {933, 6}, {1304, 11062}, {1625, 217}, {2165, 3124}, {3613, 125}, {10412, 686}}.
X(130)-cross conjugate of X(4).
X(i)-isoconjugate of X(j) for these (i,j): {{54, 811}, {75, 933}, {95, 162}, {97, 823}, {99, 2190}, {163, 276}, {275, 662}, {648, 2167}, {799, 8882}, {1969, 14586}, {2148, 6331}, {2169, 6528}, {4575, 8795}, {4592, 8884}}.
X(10632)-vertex conjugate of X(10633).
crosspoint of X(i) and X(j) for these (i,j): {{5, 1625}, {6, 933}, {112, 1173}, {523, 647}, {1304, 11079}, {6368, 12077}}.
crossdifference of every pair of points on line {2, 95}.
crosssum of X(i) and X(j) for these (i,j): {{2, 6368}, {54, 15412}, {110, 648}, {140, 525}, {9033, 14920}}.
barycentric product X(i)X(j) for these {i,j}: {{3, 12077}, {5, 647}, {6, 6368}, {48, 2618}, {51, 525}, {53, 520}, {74, 14391}, {125, 1625}, {216, 523}, {217, 850}, {265, 2081}, {311, 3049}, {343, 512}, {418, 14618}, {656, 1953}, {810, 14213}, {1154, 14582}, {1393, 8611}, {1568, 2433}, {2179, 14208}, {2501, 5562}, {2617, 3708}, {3199, 3265}, {6587, 8798}, {14575, 15415}}.
barycentric quotient X(i)/X(j) for these {i,j}: {{5, 6331}, {32, 933}, {51, 648}, {53, 6528}, {216, 99}, {217, 110}, {343, 670}, {418, 4558}, {512, 275}, {523, 276}, {647, 95}, {669, 8882}, {798, 2190}, {810, 2167}, {1953, 811}, {2081, 340}, {2179, 162}, {2181, 823}, {2489, 8884}, {2501, 8795}, {2618, 1969}, {2971, 15422}, {2972, 15414}, {3049, 54}, {3199, 107}, {5562, 4563}, {6368, 76}, {12077, 264}, {14391, 3260}, {14569, 15352}, {14575, 14586}}.
midpoint of X(4) and X(15412).
4 X[647] - X[9409].
isogonal of the isotomic of X(6368).
orthic isogonal conjugate of X(3269).
X(647)-daleth conjugate of X(3569).
X(i)-Ceva conjugate of X(j) for these (i,j): {{4, 3269}, {523, 12077}, {933, 6}, {1304, 11062}, {1625, 217}, {2165, 3124}, {3613, 125}, {10412, 686}}.
X(130)-cross conjugate of X(4).
X(i)-isoconjugate of X(j) for these (i,j): {{54, 811}, {75, 933}, {95, 162}, {97, 823}, {99, 2190}, {163, 276}, {275, 662}, {648, 2167}, {799, 8882}, {1969, 14586}, {2148, 6331}, {2169, 6528}, {4575, 8795}, {4592, 8884}}.
X(10632)-vertex conjugate of X(10633).
crosspoint of X(i) and X(j) for these (i,j): {{5, 1625}, {6, 933}, {112, 1173}, {523, 647}, {1304, 11079}, {6368, 12077}}.
crossdifference of every pair of points on line {2, 95}.
crosssum of X(i) and X(j) for these (i,j): {{2, 6368}, {54, 15412}, {110, 648}, {140, 525}, {9033, 14920}}.
barycentric product X(i)X(j) for these {i,j}: {{3, 12077}, {5, 647}, {6, 6368}, {48, 2618}, {51, 525}, {53, 520}, {74, 14391}, {125, 1625}, {216, 523}, {217, 850}, {265, 2081}, {311, 3049}, {343, 512}, {418, 14618}, {656, 1953}, {810, 14213}, {1154, 14582}, {1393, 8611}, {1568, 2433}, {2179, 14208}, {2501, 5562}, {2617, 3708}, {3199, 3265}, {6587, 8798}, {14575, 15415}}.
barycentric quotient X(i)/X(j) for these {i,j}: {{5, 6331}, {32, 933}, {51, 648}, {53, 6528}, {216, 99}, {217, 110}, {343, 670}, {418, 4558}, {512, 275}, {523, 276}, {647, 95}, {669, 8882}, {798, 2190}, {810, 2167}, {1953, 811}, {2081, 340}, {2179, 162}, {2181, 823}, {2489, 8884}, {2501, 8795}, {2618, 1969}, {2971, 15422}, {2972, 15414}, {3049, 54}, {3199, 107}, {5562, 4563}, {6368, 76}, {12077, 264}, {14391, 3260}, {14569, 15352}, {14575, 14586}}.
------------------------------ ------------------------------ -
Center: a^2 (b-c)^2 (b+c)^2 (a^2-b^2-c^2) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^8-a^6 b^2-a^4 b^4+a^2 b^6-a^6 c^2+3 a^4 b^2 c^2-3 a^2 b^4 c^2+b^6 c^2-a^4 c^4-3 a^2 b^2 c^4-2 b^4 c^4+a^2 c^6+b^2 c^6)::
on line {3269,7668}.
------------------------------ ------------------------------ -
Isogonal conjugate of line through X{2,95,97,233,275,317,577, 3087,4993,6709,8882,10311, 10313,10314,14590,14918}.
Best regards,
Peter Moses.
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