HYACINTHOS 29399

[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) =

[...]

[Peter Moses]:¨


Hi Antreas,

= a^2*(a^2 - b^2 - c^2)*(a*b + S)*(a*c + S) : :

= lies on the conics {A,B,C,X(3),X(28)}, {A,B,C,X(6),X(1805)}, {A,B,C,X(19),X(6213)}, {A,B,C,X(37),X(1123)}, {A,B,C,X(55),X(606)}, 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)
= isogonal conjugate of the anticomplement of X(13388)
= isogonal conjugate of the isotomic conjugate of X(13387)
= isogonal conjugate of the polar conjugate of X(1123)
= polar conjugate of the isotomic conjugate of X(1335)
= X(i)-Ceva conjugate of X(j) for these (i,j): {7133, 6}, {13387, 1335}
= X(i)-isoconjugate of X(j) for these (i,j): {1, 13386}, {2, 6212}, {4, 3083}, {19, 1267}, {33, 13453}, {34, 13425}, {63, 1336}, {77, 13426}, {78, 13459}, {92, 1124}, {264, 605}, {345, 13460}, {348, 13427}, {1897, 6364}, {4025, 6136}, {6213, 13424}, {13389, 14121}, {13390, 30556}
= crosspoint of X(1123) and X(13387)
= barycentric product X(i)*X(j) for these {i,j}: {1, 6213}, {3, 1123}, {4, 1335}, {6, 13387}, {19, 3084}, {25, 5391}, {77, 13456}, {78, 13438}, {92, 606}, {219, 13437}, {222, 13454}, {607, 13436}, {608, 13458}, {905, 6135}, {1659, 5414}, {1783, 6365}, {2067, 7090}, {2362, 30557}, {6413, 13457}, {7133, 13388}
= barycentric quotient X(i)/X(j) for these {i,j}: {3, 1267}, {6, 13386}, {25, 1336}, {31, 6212}, {48, 3083}, {184, 1124}, {219, 13425}, {222, 13453}, {606, 63}, {607, 13426}, {608, 13459}, {1123, 264}, {1335, 69}, {1395, 13460}, {2212, 13427}, {3084, 304}, {3937, 22107}, {5391, 305}, {6135, 6335}, {6213, 75}, {6365, 15413}, {9247, 605}, {13387, 76}, {13437, 331}, {13438, 273}, {13454, 7017}, {13456, 318}, {22383, 6364}

Best regards,
Peter.