Σάββατο, 3 Δεκεμβρίου 2011

Nice, if true!


1. Let ABC be a triangle.


Denote:

(O1), (O2), (O3) the reflections of the circumcircle (O) in AI, BI, CI, rersp.

L1, L2, L3 the radical axes of [(O2), (O3)], [(O3), (O1)], [(O1), (O2)], resp.

M1, M2, M3 the reflections of L1, L2, L3 in BC, CA, AB, resp.

Are the lines M1 ,M2, M3 concurrent? Point?


2. Let ABC be a triangle.


Denote:

(O1), (O2), (O3) the reflections of the circumcircle (O) in AI, BI, CI, rersp.

(O12), (O13) the reflections of (O1) in BI, CI, resp.

(O23), (O21) the reflections of (O2) in CI, AI, resp.

(O31), (O32) the reflections of (O3) in AI, BI, resp.

L1, L2, L3 the radical axes of [(O12), (O13)], [(O23), (O21)], [(O31), (O32)], resp.

M1, M2, M3 the reflections of L1, L2, L3 in BC, CA, AB, resp.

Are the lines M1 ,M2, M3 concurrent? Point?

Addendum: Loci ?? (P instead of I)

APH, 3 December 2011

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1. P=(x:y:z) can be any point.

The point of concurrence is:
{a^2 (b^2 c^2 x^2 + a^2 c^2 x y - c^4 x y + a^2 b^2 x z - b^4 x z +
a^4 y z - a^2 b^2 y z - a^2 c^2 y z),
b^2 (b^2 c^2 x y - c^4 x y + a^2 c^2 y^2 - a^2 b^2 x z + b^4 x z -
b^2 c^2 x z - a^4 y z + a^2 b^2 y z),
c^2 (-a^2 c^2 x y - b^2 c^2 x y + c^4 x y - b^4 x z + b^2 c^2 x z -
a^4 y z + a^2 c^2 y z + a^2 b^2 z^2)}

2. The locus of P is circumcircle, line at infinity and the curve:

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

through X1, X3, X6

Francisco Javier García Capitán
3 December 2011

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This is the Euler-Morley quartic Q002.

Bernard Gibert
4 December 2011


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