Τρίτη 29 Οκτωβρίου 2019

HYACINTHOS 28988

[Kadir Altintas - Ercole Suppa]: 

Let ABC be a triangle, P a point and  DEF the cevian triangle of P
Let A' be the second intersection of AD with the circle (DEF)
The circle with diameter PA' intersects the circle (DEF) again at  A''
Define B'',C'' cyclically
Find the locus of points P such that triangles A''B''C'' and ABC are perspective.
 
Romantics of Geometry, problem 2970
 
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[Ercole Suppa]
 
The locus of point P(x;y:z) such that triangles A''B''C'' and ABC are perspective is Γ = {sidelines} U {Lucas cubic K007} U {Yiu quintic Q006} U {circumsextics: q1,q2,q3} , where: 
 
K007 Lucas cubic: ∑ yz(SC y -SB z)=0 
 
Q006 Yiu quintic:  ∑ (2*SA x^3 y z + c^2 x^2 y^2 z + a^2 y^3 z^2 + a^2 y^2 z^3) = 0
 
q1: a^2 c^2 x^4 y^2-b^2 c^2 x^4 y^2-c^4 x^4 y^2+a^2 c^2 x^3 y^3-b^2 c^2 x^3 y^3+c^4 x^3 y^3-a^4 x^4 y z+2 a^2 b^2 x^4 y z-b^4 x^4 y z+2 a^2 c^2 x^4 y z-2 b^2 c^2 x^4 y z-c^4 x^4 y z-a^2 c^2 x^3 y^2 z-b^2 c^2 x^3 y^2 z+c^4 x^3 y^2 z+a^4 x^2 y^3 z-2 a^2 b^2 x^2 y^3 z+b^4 x^2 y^3 z-2 b^2 c^2 x^2 y^3 z+c^4 x^2 y^3 z+a^2 b^2 x^4 z^2-b^4 x^4 z^2-b^2 c^2 x^4 z^2-a^2 b^2 x^3 y z^2+b^4 x^3 y z^2-b^2 c^2 x^3 y z^2-a^4 x^2 y^2 z^2-a^2 b^2 x^2 y^2 z^2+2 b^4 x^2 y^2 z^2-a^2 c^2 x^2 y^2 z^2-4 b^2 c^2 x^2 y^2 z^2+2 c^4 x^2 y^2 z^2-a^4 x y^3 z^2+a^2 b^2 x y^3 z^2-a^2 c^2 x y^3 z^2+a^2 b^2 x^3 z^3+b^4 x^3 z^3-b^2 c^2 x^3 z^3+a^4 x^2 y z^3+b^4 x^2 y z^3-2 a^2 c^2 x^2 y z^3-2 b^2 c^2 x^2 y z^3+c^4 x^2 y z^3-a^4 x y^2 z^3-a^2 b^2 x y^2 z^3+a^2 c^2 x y^2 z^3-2 a^4 y^3 z^3 = 0
 
q2: a^2 c^2 x^3 y^3-b^2 c^2 x^3 y^3-c^4 x^3 y^3+a^2 c^2 x^2 y^4-b^2 c^2 x^2 y^4+c^4 x^2 y^4-a^4 x^3 y^2 z+2 a^2 b^2 x^3 y^2 z-b^4 x^3 y^2 z+2 a^2 c^2 x^3 y^2 z-c^4 x^3 y^2 z+a^2 c^2 x^2 y^3 z+b^2 c^2 x^2 y^3 z-c^4 x^2 y^3 z+a^4 x y^4 z-2 a^2 b^2 x y^4 z+b^4 x y^4 z+2 a^2 c^2 x y^4 z-2 b^2 c^2 x y^4 z+c^4 x y^4 z-a^2 b^2 x^3 y z^2+b^4 x^3 y z^2+b^2 c^2 x^3 y z^2-2 a^4 x^2 y^2 z^2+a^2 b^2 x^2 y^2 z^2+b^4 x^2 y^2 z^2+4 a^2 c^2 x^2 y^2 z^2+b^2 c^2 x^2 y^2 z^2-2 c^4 x^2 y^2 z^2-a^4 x y^3 z^2+a^2 b^2 x y^3 z^2+a^2 c^2 x y^3 z^2+a^4 y^4 z^2-a^2 b^2 y^4 z^2+a^2 c^2 y^4 z^2+2 b^4 x^3 z^3+a^2 b^2 x^2 y z^3+b^4 x^2 y z^3-b^2 c^2 x^2 y z^3-a^4 x y^2 z^3-b^4 x y^2 z^3+2 a^2 c^2 x y^2 z^3+2 b^2 c^2 x y^2 z^3-c^4 x y^2 z^3-a^4 y^3 z^3-a^2 b^2 y^3 z^3+a^2 c^2 y^3 z^3 = 0
 
q3: 2 c^4 x^3 y^3-a^2 c^2 x^3 y^2 z+b^2 c^2 x^3 y^2 z+c^4 x^3 y^2 z+a^2 c^2 x^2 y^3 z-b^2 c^2 x^2 y^3 z+c^4 x^2 y^3 z-a^4 x^3 y z^2+2 a^2 b^2 x^3 y z^2-b^4 x^3 y z^2+2 a^2 c^2 x^3 y z^2-c^4 x^3 y z^2-2 a^4 x^2 y^2 z^2+4 a^2 b^2 x^2 y^2 z^2-2 b^4 x^2 y^2 z^2+a^2 c^2 x^2 y^2 z^2+b^2 c^2 x^2 y^2 z^2+c^4 x^2 y^2 z^2-a^4 x y^3 z^2+2 a^2 b^2 x y^3 z^2-b^4 x y^3 z^2+2 b^2 c^2 x y^3 z^2-c^4 x y^3 z^2+a^2 b^2 x^3 z^3-b^4 x^3 z^3-b^2 c^2 x^3 z^3+a^2 b^2 x^2 y z^3-b^4 x^2 y z^3+b^2 c^2 x^2 y z^3-a^4 x y^2 z^3+a^2 b^2 x y^2 z^3+a^2 c^2 x y^2 z^3-a^4 y^3 z^3+a^2 b^2 y^3 z^3-a^2 c^2 y^3 z^3+a^2 b^2 x^2 z^4+b^4 x^2 z^4-b^2 c^2 x^2 z^4+a^4 x y z^4+2 a^2 b^2 x y z^4+b^4 x y z^4-2 a^2 c^2 x y z^4-2 b^2 c^2 x y z^4+c^4 x y z^4+a^4 y^2 z^4+a^2 b^2 y^2 z^4-a^2 c^2 y^2 z^4 = 0
 
 
** Locus Γ passes through X(i) for these i: {2,4,7,8,20,69,189,253,329,1032,1034,5932,14361,14362,14365}
 
 
** Let Q=Q(P) be the concurrency point of AA'', BB'', CC''. Pairs {P=X(i),Q=X(j)} for these {i,j}: {4, 6526}
 
 
** Some points:
 
 
Q = X(2)X(12505) ∩ X(67)X(10354)
 
= (a^2+b^2-5 c^2) (a^2-5 b^2+c^2) (a^4-4 a^2 b^2+b^4-c^4) (a^4-b^4-4 a^2 c^2+c^4) :: (barys)
 
= (648 R^4-162 R^2 SW+9 SW^2)S^4 + (-54 R^2 SB SC SW+9 SB SC SW^2-24 R^2 SW^3-3 SB SW^3-3 SC SW^3+6 SW^4)S^2 + 2 SB SC SW^4  :: (barys)
 
= lies on these lines: {2,12505}, {67,10354}, {599,5486}, {858,14262}, {1296,16063}, {5094,21448}, {5485,16051}
= barycentric product of X(i) and X(j) for these {i,j}: {338,15406}, {5485,5486}
 
= barycentric quotient of X(i) and X(j) for these {i,j}: {115,5512}, {5485,11185}, {5486,1992}, {15406,249}, {21448,1995}
 
= trilinear product of X(i) and X(j) for these {i,j}: {1109,15406}, {1109,15406}
 
= trilinear quotient of X(i) and X(j) for these {i,j}: {1109, 5512}
 
= (6-9-13) search numbers [-1.29187724958749794, 6.06644025118795909, 0.0370722693561757046]
 
 
 
Best regards
Ercole Suppa
 

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