Τρίτη, 20 Δεκεμβρίου 2011

A "COLOR" THEOREM

Let ABC be a triangle, P a point, A'B'C' the cevian triangle of P and X,Y two fixed points.

From A' we draw two RED lines joining it with X,Y
From B' we draw two BLUE lines joining it with X,Y
From C' we draw two GREEN lines joining it with X,Y.

The lines:

Line joining the intersections of RED with BLUE lines other than X,Y
Line joining the intersections of BLUE with GREEN lines other than X,Y
Line joining the intersections of GREEN with RED lines other than X,Y

are concurrent.

The problem without colors is:

Let ABC be a triangle, P a point, A'B'C' the cevian triangle of P
and X,Y two points.

The lines:

(A'X /\ B'Y) \/ (A'Y /\ B'X)

(B'X /\ C'Y) \/ (B'Y /\ C'X)

(C'X /\ A'Y) \/ (C'Y /\ A'X)

are concurrent.

Proof ??

APH 20 December 2011

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For P=(u:v:w), X=(u1:v1:w1), Y=(u2:v2:w2) the point of concurrence is:

(u2 w1^2 w2 x^3 y^3 + u1 w1 w2^2 x^3 y^3 + u2 v2 w1^2 x^3 y^2 z -
2 u2 v1 w1 w2 x^3 y^2 z - 2 u1 v2 w1 w2 x^3 y^2 z +
u1 v1 w2^2 x^3 y^2 z - u2^2 w1^2 x^2 y^3 z - u1^2 w2^2 x^2 y^3 z -
2 u2 v1 v2 w1 x^3 y z^2 + u1 v2^2 w1 x^3 y z^2 +
u2 v1^2 w2 x^3 y z^2 - 2 u1 v1 v2 w2 x^3 y z^2 +
2 u2^2 v1 w1 x^2 y^2 z^2 + 2 u1^2 v2 w2 x^2 y^2 z^2 -
u1 u2^2 w1 x y^3 z^2 - u1^2 u2 w2 x y^3 z^2 + u2 v1^2 v2 x^3 z^3 +
u1 v1 v2^2 x^3 z^3 - u2^2 v1^2 x^2 y z^3 - u1^2 v2^2 x^2 y z^3 -
u1 u2^2 v1 x y^2 z^3 - u1^2 u2 v2 x y^2 z^3 + 2 u1^2 u2^2 y^3 z^3 :
v2 w1^2 w2 x^3 y^3 + v1 w1 w2^2 x^3 y^3 - v2^2 w1^2 x^3 y^2 z -
v1^2 w2^2 x^3 y^2 z + u2 v2 w1^2 x^2 y^3 z -
2 u2 v1 w1 w2 x^2 y^3 z - 2 u1 v2 w1 w2 x^2 y^3 z +
u1 v1 w2^2 x^2 y^3 z - v1 v2^2 w1 x^3 y z^2 -
v1^2 v2 w2 x^3 y z^2 + 2 u1 v2^2 w1 x^2 y^2 z^2 +
2 u2 v1^2 w2 x^2 y^2 z^2 + u2^2 v1 w1 x y^3 z^2 -
2 u1 u2 v2 w1 x y^3 z^2 - 2 u1 u2 v1 w2 x y^3 z^2 +
u1^2 v2 w2 x y^3 z^2 + 2 v1^2 v2^2 x^3 z^3 - u2 v1^2 v2 x^2 y z^3 -
u1 v1 v2^2 x^2 y z^3 - u2^2 v1^2 x y^2 z^3 - u1^2 v2^2 x y^2 z^3 +
u1 u2^2 v1 y^3 z^3 + u1^2 u2 v2 y^3 z^3 :
2 w1^2 w2^2 x^3 y^3 - v2 w1^2 w2 x^3 y^2 z - v1 w1 w2^2 x^3 y^2 z -
u2 w1^2 w2 x^2 y^3 z - u1 w1 w2^2 x^2 y^3 z - v2^2 w1^2 x^3 y z^2 -
v1^2 w2^2 x^3 y z^2 + 2 u2 v2 w1^2 x^2 y^2 z^2 +
2 u1 v1 w2^2 x^2 y^2 z^2 - u2^2 w1^2 x y^3 z^2 -
u1^2 w2^2 x y^3 z^2 + v1 v2^2 w1 x^3 z^3 + v1^2 v2 w2 x^3 z^3 -
2 u2 v1 v2 w1 x^2 y z^3 + u1 v2^2 w1 x^2 y z^3 +
u2 v1^2 w2 x^2 y z^3 - 2 u1 v1 v2 w2 x^2 y z^3 +
u2^2 v1 w1 x y^2 z^3 - 2 u1 u2 v2 w1 x y^2 z^3 -
2 u1 u2 v1 w2 x y^2 z^3 + u1^2 v2 w2 x y^2 z^3 +
u1 u2^2 w1 y^3 z^3 + u1^2 u2 w2 y^3 z^3).

Francisco Javier García Capitán
21 December 2011