HYACINTHOS 29397

[Ercole Suppa]

Let ABC be a triangle.

Denote:

ω = the circumcircle of ABC

ω_a,  ω_b, ω_c = the circles with diameters BC, CA, AB, resp.

Γ_a = the circle internally tangent to ω, ω_b, ω_c

A' = the touch point of ω and Γ_a.

Define B',C' cyclically

Prove that:

(1) The lines AA', BB', CC' concur at a point X

(2) Which is the point X in triangle ABC ?

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[Ercole Suppa]

X = ISOGONAL CONJUGATE OF X(13386) =

= a^2(a^2+b^2-c^2)(a^2-b^2+c^2)(a^4 b-2 a^2 b^3+b^5+2 a^3 b c-2 a b^3 c-4 a^2 b c^2-4 b^3 c^2-2 a b c^3+3 b c^4-2 (a^3-a b^2-a^2 c-3 b^2 c-a c^2+c^3) S)(a^5-4 a^3 b^2+3 a b^4-2 a^3 b c-2 a b^3 c-2 a^3 c^2-4 a b^2 c^2+2 a b c^3+a c^4+2 (3 a^2 b-b^3+a^2 c+b^2 c+b c^2-c^3) S) : : (barys)

= lies on the conic {A,B,C,X(3),X(28)}, the cubic K171 and these lines: {3,6213}, {4,16027}, {19,25}, {28,1123}, {41,5416}, {48,5414}, {56,2362}, {104,6135}, {480,11498}, {603,606}, {607,5413}, {608,5412}, {958,7090}, {1335,1437}, {1400,5415}, {1436,13456}, {1444,5391}, {1598,6212}, {1633,30296}, {1659,13887}, {1950,26953}, {2066,2183}, {2082,19005}, {2285,19006}, {2333,8948}, {3069,30385}, {3207,30336}, {3542,16033}, {6204,13889}, {7289,13388}, {7348,13943}, {16036,19173}, {19215,19588}

 

= isogonal conjugate of X(13386)

= (6-9-13)  search numbers:  [9.1774451739764907393, 10.4588492924639740342, -7.8358212624029063387]


Best regards
Ercole Suppa