Δευτέρα, 5 Δεκεμβρίου 2011

Reflections of the Circumcircle in the cevians

Let ABC be a triangle and P a point. Let (O1),(O2), (O3) be the reflections of the circumcircle (O) in BC, CA, AB, resp.

Let (O11), (O22), (O33) be the reflections of (O1), (O2), (O3) in AP, BP, CP, resp., L1,L2,L3 the radical axes of [(O22),(O33)], [(O33),(O11)], [(O11), (O22)], resp. and M1,M2,M3 the parallels to L1,L2,L3 through A,B,C resp.

Which is the locus of P such that M1,M2,M3 are concurrent?

APH, 5 December 2011

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Locus:

(-a - b + c) (a - b + c) (-a + b + c) (a + b + c) (x + y +
z) (-a^4 c^4 x^3 y^2 + a^2 b^2 c^4 x^3 y^2 + 2 a^2 c^6 x^3 y^2 +
b^2 c^6 x^3 y^2 - c^8 x^3 y^2 - a^2 b^2 c^4 x^2 y^3 +
b^4 c^4 x^2 y^3 - a^2 c^6 x^2 y^3 - 2 b^2 c^6 x^2 y^3 +
c^8 x^2 y^3 - a^6 b^2 x^3 y z + 3 a^4 b^4 x^3 y z -
3 a^2 b^6 x^3 y z + b^8 x^3 y z + a^6 c^2 x^3 y z -
b^6 c^2 x^3 y z - 3 a^4 c^4 x^3 y z + 3 a^2 c^6 x^3 y z +
b^2 c^6 x^3 y z - c^8 x^3 y z - a^8 x^2 y^2 z +
2 a^6 b^2 x^2 y^2 z - 2 a^2 b^6 x^2 y^2 z + b^8 x^2 y^2 z +
2 a^6 c^2 x^2 y^2 z - 2 b^6 c^2 x^2 y^2 z - a^4 c^4 x^2 y^2 z +
b^4 c^4 x^2 y^2 z - a^8 x y^3 z + 3 a^6 b^2 x y^3 z -
3 a^4 b^4 x y^3 z + a^2 b^6 x y^3 z + a^6 c^2 x y^3 z -
b^6 c^2 x y^3 z + 3 b^4 c^4 x y^3 z - a^2 c^6 x y^3 z -
3 b^2 c^6 x y^3 z + c^8 x y^3 z + a^4 b^4 x^3 z^2 -
2 a^2 b^6 x^3 z^2 + b^8 x^3 z^2 - a^2 b^4 c^2 x^3 z^2 -
b^6 c^2 x^3 z^2 + a^8 x^2 y z^2 - 2 a^6 b^2 x^2 y z^2 +
a^4 b^4 x^2 y z^2 - 2 a^6 c^2 x^2 y z^2 - b^4 c^4 x^2 y z^2 +
2 a^2 c^6 x^2 y z^2 + 2 b^2 c^6 x^2 y z^2 - c^8 x^2 y z^2 -
a^4 b^4 x y^2 z^2 + 2 a^2 b^6 x y^2 z^2 - b^8 x y^2 z^2 +
2 b^6 c^2 x y^2 z^2 + a^4 c^4 x y^2 z^2 - 2 a^2 c^6 x y^2 z^2 -
2 b^2 c^6 x y^2 z^2 + c^8 x y^2 z^2 - a^8 y^3 z^2 +
2 a^6 b^2 y^3 z^2 - a^4 b^4 y^3 z^2 + a^6 c^2 y^3 z^2 +
a^4 b^2 c^2 y^3 z^2 + a^2 b^6 x^2 z^3 - b^8 x^2 z^3 +
a^2 b^4 c^2 x^2 z^3 + 2 b^6 c^2 x^2 z^3 - b^4 c^4 x^2 z^3 +
a^8 x y z^3 - a^6 b^2 x y z^3 + a^2 b^6 x y z^3 - b^8 x y z^3 -
3 a^6 c^2 x y z^3 + 3 b^6 c^2 x y z^3 + 3 a^4 c^4 x y z^3 -
3 b^4 c^4 x y z^3 - a^2 c^6 x y z^3 + b^2 c^6 x y z^3 +
a^8 y^2 z^3 - a^6 b^2 y^2 z^3 - 2 a^6 c^2 y^2 z^3 -
a^4 b^2 c^2 y^2 z^3 + a^4 c^4 y^2 z^3) (-a^2 c^4 x^4 y^2 +
b^2 c^4 x^4 y^2 + c^6 x^4 y^2 - c^6 x^3 y^3 + a^2 c^4 x^2 y^4 -
b^2 c^4 x^2 y^4 + c^6 x^2 y^4 + a^6 x^4 y z - 2 a^4 b^2 x^4 y z +
a^2 b^4 x^4 y z - 2 a^4 c^2 x^4 y z + 2 b^4 c^2 x^4 y z +
a^2 c^4 x^4 y z + 2 b^2 c^4 x^4 y z + 2 a^6 x^3 y^2 z -
3 a^4 b^2 x^3 y^2 z + b^6 x^3 y^2 z - 4 a^4 c^2 x^3 y^2 z +
4 a^2 b^2 c^2 x^3 y^2 z - 6 b^2 c^4 x^3 y^2 z + 2 c^6 x^3 y^2 z +
a^6 x^2 y^3 z - 3 a^2 b^4 x^2 y^3 z + 2 b^6 x^2 y^3 z +
4 a^2 b^2 c^2 x^2 y^3 z - 4 b^4 c^2 x^2 y^3 z -
6 a^2 c^4 x^2 y^3 z + 2 c^6 x^2 y^3 z + a^4 b^2 x y^4 z -
2 a^2 b^4 x y^4 z + b^6 x y^4 z + 2 a^4 c^2 x y^4 z -
2 b^4 c^2 x y^4 z + 2 a^2 c^4 x y^4 z + b^2 c^4 x y^4 z -
a^2 b^4 x^4 z^2 + b^6 x^4 z^2 + b^4 c^2 x^4 z^2 +
2 a^6 x^3 y z^2 - 4 a^4 b^2 x^3 y z^2 + 2 b^6 x^3 y z^2 -
3 a^4 c^2 x^3 y z^2 + 4 a^2 b^2 c^2 x^3 y z^2 -
6 b^4 c^2 x^3 y z^2 + c^6 x^3 y z^2 + 3 a^6 x^2 y^2 z^2 -
3 a^4 b^2 x^2 y^2 z^2 - 3 a^2 b^4 x^2 y^2 z^2 +
3 b^6 x^2 y^2 z^2 - 3 a^4 c^2 x^2 y^2 z^2 -
6 a^2 b^2 c^2 x^2 y^2 z^2 - 3 b^4 c^2 x^2 y^2 z^2 -
3 a^2 c^4 x^2 y^2 z^2 - 3 b^2 c^4 x^2 y^2 z^2 +
3 c^6 x^2 y^2 z^2 + 2 a^6 x y^3 z^2 - 4 a^2 b^4 x y^3 z^2 +
2 b^6 x y^3 z^2 - 6 a^4 c^2 x y^3 z^2 + 4 a^2 b^2 c^2 x y^3 z^2 -
3 b^4 c^2 x y^3 z^2 + c^6 x y^3 z^2 + a^6 y^4 z^2 -
a^4 b^2 y^4 z^2 + a^4 c^2 y^4 z^2 - b^6 x^3 z^3 + a^6 x^2 y z^3 -
6 a^2 b^4 x^2 y z^3 + 2 b^6 x^2 y z^3 + 4 a^2 b^2 c^2 x^2 y z^3 -
3 a^2 c^4 x^2 y z^3 - 4 b^2 c^4 x^2 y z^3 + 2 c^6 x^2 y z^3 +
2 a^6 x y^2 z^3 - 6 a^4 b^2 x y^2 z^3 + b^6 x y^2 z^3 +
4 a^2 b^2 c^2 x y^2 z^3 - 4 a^2 c^4 x y^2 z^3 -
3 b^2 c^4 x y^2 z^3 + 2 c^6 x y^2 z^3 - a^6 y^3 z^3 +
a^2 b^4 x^2 z^4 + b^6 x^2 z^4 - b^4 c^2 x^2 z^4 +
2 a^4 b^2 x y z^4 + 2 a^2 b^4 x y z^4 + a^4 c^2 x y z^4 +
b^4 c^2 x y z^4 - 2 a^2 c^4 x y z^4 - 2 b^2 c^4 x y z^4 +
c^6 x y z^4 + a^6 y^2 z^4 + a^4 b^2 y^2 z^4 - a^4 c^2 y^2 z^4)

Francisco Javier García Capitán
5 December 2011

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