NPCs and Radical Axes

Let ABC be a triangle, P a point, A1B1C1 and A2B2C2 its cevian and cyclocevian triangles, resp.

Denote:

R1 := the radical axis of the NPCs of A1B2C2, A2B1C1

R2 := the radical axis of the NPCs of B1C2A2, B2C1A1

R3 := the radical axis of the NPCs of C1A2B2, C2A1B1

The lines R1,R2,R3 are concurrent.
(and also the radical axis of the NPCs of A1B1C1,A2B2C2)

Point of concurrence?

Variation:

-- A'B'C' = Circumcevian triangle of point P wrt ABC.

Generalization:

-- A1B1C1, A2B2C2 = two triangles inscribed in the same circle.

APH, 14 December 2011

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For the triangle case [cyclocevian] this is the point:


{a^2 (-a^4 c^4 u^3 v^3 + b^4 c^4 u^3 v^3 + 2 a^2 c^6 u^3 v^3 -
c^8 u^3 v^3 + a^6 c^2 u^3 v^2 w + 2 a^4 b^2 c^2 u^3 v^2 w -
7 a^2 b^4 c^2 u^3 v^2 w + 4 b^6 c^2 u^3 v^2 w -
3 a^4 c^4 u^3 v^2 w - 6 a^2 b^2 c^4 u^3 v^2 w -
7 b^4 c^4 u^3 v^2 w + 3 a^2 c^6 u^3 v^2 w + 4 b^2 c^6 u^3 v^2 w -
c^8 u^3 v^2 w + a^6 c^2 u^2 v^3 w - 3 a^2 b^4 c^2 u^2 v^3 w +
2 b^6 c^2 u^2 v^3 w - 4 a^4 c^4 u^2 v^3 w -
6 a^2 b^2 c^4 u^2 v^3 w - 6 b^4 c^4 u^2 v^3 w +
5 a^2 c^6 u^2 v^3 w + 6 b^2 c^6 u^2 v^3 w - 2 c^8 u^2 v^3 w +
a^6 b^2 u^3 v w^2 - 3 a^4 b^4 u^3 v w^2 + 3 a^2 b^6 u^3 v w^2 -
b^8 u^3 v w^2 + 2 a^4 b^2 c^2 u^3 v w^2 -
6 a^2 b^4 c^2 u^3 v w^2 + 4 b^6 c^2 u^3 v w^2 -
7 a^2 b^2 c^4 u^3 v w^2 - 7 b^4 c^4 u^3 v w^2 +
4 b^2 c^6 u^3 v w^2 + 2 a^6 b^2 u^2 v^2 w^2 -
6 a^4 b^4 u^2 v^2 w^2 + 6 a^2 b^6 u^2 v^2 w^2 -
2 b^8 u^2 v^2 w^2 + 2 a^6 c^2 u^2 v^2 w^2 -
10 a^2 b^4 c^2 u^2 v^2 w^2 + 8 b^6 c^2 u^2 v^2 w^2 -
6 a^4 c^4 u^2 v^2 w^2 - 10 a^2 b^2 c^4 u^2 v^2 w^2 -
12 b^4 c^4 u^2 v^2 w^2 + 6 a^2 c^6 u^2 v^2 w^2 +
8 b^2 c^6 u^2 v^2 w^2 - 2 c^8 u^2 v^2 w^2 + a^6 b^2 u v^3 w^2 -
3 a^4 b^4 u v^3 w^2 + 3 a^2 b^6 u v^3 w^2 - b^8 u v^3 w^2 +
2 a^6 c^2 u v^3 w^2 - 2 a^4 b^2 c^2 u v^3 w^2 -
4 a^2 b^4 c^2 u v^3 w^2 + 4 b^6 c^2 u v^3 w^2 -
5 a^4 c^4 u v^3 w^2 - 3 a^2 b^2 c^4 u v^3 w^2 -
6 b^4 c^4 u v^3 w^2 + 4 a^2 c^6 u v^3 w^2 + 4 b^2 c^6 u v^3 w^2 -
c^8 u v^3 w^2 - a^4 b^4 u^3 w^3 + 2 a^2 b^6 u^3 w^3 -
b^8 u^3 w^3 + b^4 c^4 u^3 w^3 + a^6 b^2 u^2 v w^3 -
4 a^4 b^4 u^2 v w^3 + 5 a^2 b^6 u^2 v w^3 - 2 b^8 u^2 v w^3 -
6 a^2 b^4 c^2 u^2 v w^3 + 6 b^6 c^2 u^2 v w^3 -
3 a^2 b^2 c^4 u^2 v w^3 - 6 b^4 c^4 u^2 v w^3 +
2 b^2 c^6 u^2 v w^3 + 2 a^6 b^2 u v^2 w^3 - 5 a^4 b^4 u v^2 w^3 +
4 a^2 b^6 u v^2 w^3 - b^8 u v^2 w^3 + a^6 c^2 u v^2 w^3 -
2 a^4 b^2 c^2 u v^2 w^3 - 3 a^2 b^4 c^2 u v^2 w^3 +
4 b^6 c^2 u v^2 w^3 - 3 a^4 c^4 u v^2 w^3 -
4 a^2 b^2 c^4 u v^2 w^3 - 6 b^4 c^4 u v^2 w^3 +
3 a^2 c^6 u v^2 w^3 + 4 b^2 c^6 u v^2 w^3 - c^8 u v^2 w^3 +
a^6 b^2 v^3 w^3 - 2 a^4 b^4 v^3 w^3 + a^2 b^6 v^3 w^3 +
a^6 c^2 v^3 w^3 - a^2 b^4 c^2 v^3 w^3 - 2 a^4 c^4 v^3 w^3 -
a^2 b^2 c^4 v^3 w^3 + a^2 c^6 v^3 w^3),
b^2 (a^4 c^4 u^3 v^3 - b^4 c^4 u^3 v^3 + 2 b^2 c^6 u^3 v^3 -
c^8 u^3 v^3 + 2 a^6 c^2 u^3 v^2 w - 3 a^4 b^2 c^2 u^3 v^2 w +
b^6 c^2 u^3 v^2 w - 6 a^4 c^4 u^3 v^2 w -
6 a^2 b^2 c^4 u^3 v^2 w - 4 b^4 c^4 u^3 v^2 w +
6 a^2 c^6 u^3 v^2 w + 5 b^2 c^6 u^3 v^2 w - 2 c^8 u^3 v^2 w +
4 a^6 c^2 u^2 v^3 w - 7 a^4 b^2 c^2 u^2 v^3 w +
2 a^2 b^4 c^2 u^2 v^3 w + b^6 c^2 u^2 v^3 w -
7 a^4 c^4 u^2 v^3 w - 6 a^2 b^2 c^4 u^2 v^3 w -
3 b^4 c^4 u^2 v^3 w + 4 a^2 c^6 u^2 v^3 w + 3 b^2 c^6 u^2 v^3 w -
c^8 u^2 v^3 w - a^8 u^3 v w^2 + 3 a^6 b^2 u^3 v w^2 -
3 a^4 b^4 u^3 v w^2 + a^2 b^6 u^3 v w^2 + 4 a^6 c^2 u^3 v w^2 -
4 a^4 b^2 c^2 u^3 v w^2 - 2 a^2 b^4 c^2 u^3 v w^2 +
2 b^6 c^2 u^3 v w^2 - 6 a^4 c^4 u^3 v w^2 -
3 a^2 b^2 c^4 u^3 v w^2 - 5 b^4 c^4 u^3 v w^2 +
4 a^2 c^6 u^3 v w^2 + 4 b^2 c^6 u^3 v w^2 - c^8 u^3 v w^2 -
2 a^8 u^2 v^2 w^2 + 6 a^6 b^2 u^2 v^2 w^2 -
6 a^4 b^4 u^2 v^2 w^2 + 2 a^2 b^6 u^2 v^2 w^2 +
8 a^6 c^2 u^2 v^2 w^2 - 10 a^4 b^2 c^2 u^2 v^2 w^2 +
2 b^6 c^2 u^2 v^2 w^2 - 12 a^4 c^4 u^2 v^2 w^2 -
10 a^2 b^2 c^4 u^2 v^2 w^2 - 6 b^4 c^4 u^2 v^2 w^2 +
8 a^2 c^6 u^2 v^2 w^2 + 6 b^2 c^6 u^2 v^2 w^2 -
2 c^8 u^2 v^2 w^2 - a^8 u v^3 w^2 + 3 a^6 b^2 u v^3 w^2 -
3 a^4 b^4 u v^3 w^2 + a^2 b^6 u v^3 w^2 + 4 a^6 c^2 u v^3 w^2 -
6 a^4 b^2 c^2 u v^3 w^2 + 2 a^2 b^4 c^2 u v^3 w^2 -
7 a^4 c^4 u v^3 w^2 - 7 a^2 b^2 c^4 u v^3 w^2 +
4 a^2 c^6 u v^3 w^2 + a^6 b^2 u^3 w^3 - 2 a^4 b^4 u^3 w^3 +
a^2 b^6 u^3 w^3 - a^4 b^2 c^2 u^3 w^3 + b^6 c^2 u^3 w^3 -
a^2 b^2 c^4 u^3 w^3 - 2 b^4 c^4 u^3 w^3 + b^2 c^6 u^3 w^3 -
a^8 u^2 v w^3 + 4 a^6 b^2 u^2 v w^3 - 5 a^4 b^4 u^2 v w^3 +
2 a^2 b^6 u^2 v w^3 + 4 a^6 c^2 u^2 v w^3 -
3 a^4 b^2 c^2 u^2 v w^3 - 2 a^2 b^4 c^2 u^2 v w^3 +
b^6 c^2 u^2 v w^3 - 6 a^4 c^4 u^2 v w^3 -
4 a^2 b^2 c^4 u^2 v w^3 - 3 b^4 c^4 u^2 v w^3 +
4 a^2 c^6 u^2 v w^3 + 3 b^2 c^6 u^2 v w^3 - c^8 u^2 v w^3 -
2 a^8 u v^2 w^3 + 5 a^6 b^2 u v^2 w^3 - 4 a^4 b^4 u v^2 w^3 +
a^2 b^6 u v^2 w^3 + 6 a^6 c^2 u v^2 w^3 -
6 a^4 b^2 c^2 u v^2 w^3 - 6 a^4 c^4 u v^2 w^3 -
3 a^2 b^2 c^4 u v^2 w^3 + 2 a^2 c^6 u v^2 w^3 - a^8 v^3 w^3 +
2 a^6 b^2 v^3 w^3 - a^4 b^4 v^3 w^3 + a^4 c^4 v^3 w^3),
c^2 (a^6 c^2 u^3 v^3 - a^4 b^2 c^2 u^3 v^3 - a^2 b^4 c^2 u^3 v^3 +
b^6 c^2 u^3 v^3 - 2 a^4 c^4 u^3 v^3 - 2 b^4 c^4 u^3 v^3 +
a^2 c^6 u^3 v^3 + b^2 c^6 u^3 v^3 - a^8 u^3 v^2 w +
4 a^6 b^2 u^3 v^2 w - 6 a^4 b^4 u^3 v^2 w + 4 a^2 b^6 u^3 v^2 w -
b^8 u^3 v^2 w + 3 a^6 c^2 u^3 v^2 w - 4 a^4 b^2 c^2 u^3 v^2 w -
3 a^2 b^4 c^2 u^3 v^2 w + 4 b^6 c^2 u^3 v^2 w -
3 a^4 c^4 u^3 v^2 w - 2 a^2 b^2 c^4 u^3 v^2 w -
5 b^4 c^4 u^3 v^2 w + a^2 c^6 u^3 v^2 w + 2 b^2 c^6 u^3 v^2 w -
a^8 u^2 v^3 w + 4 a^6 b^2 u^2 v^3 w - 6 a^4 b^4 u^2 v^3 w +
4 a^2 b^6 u^2 v^3 w - b^8 u^2 v^3 w + 4 a^6 c^2 u^2 v^3 w -
3 a^4 b^2 c^2 u^2 v^3 w - 4 a^2 b^4 c^2 u^2 v^3 w +
3 b^6 c^2 u^2 v^3 w - 5 a^4 c^4 u^2 v^3 w -
2 a^2 b^2 c^4 u^2 v^3 w - 3 b^4 c^4 u^2 v^3 w +
2 a^2 c^6 u^2 v^3 w + b^2 c^6 u^2 v^3 w + 2 a^6 b^2 u^3 v w^2 -
6 a^4 b^4 u^3 v w^2 + 6 a^2 b^6 u^3 v w^2 - 2 b^8 u^3 v w^2 -
3 a^4 b^2 c^2 u^3 v w^2 - 6 a^2 b^4 c^2 u^3 v w^2 +
5 b^6 c^2 u^3 v w^2 - 4 b^4 c^4 u^3 v w^2 + b^2 c^6 u^3 v w^2 -
2 a^8 u^2 v^2 w^2 + 8 a^6 b^2 u^2 v^2 w^2 -
12 a^4 b^4 u^2 v^2 w^2 + 8 a^2 b^6 u^2 v^2 w^2 -
2 b^8 u^2 v^2 w^2 + 6 a^6 c^2 u^2 v^2 w^2 -
10 a^4 b^2 c^2 u^2 v^2 w^2 - 10 a^2 b^4 c^2 u^2 v^2 w^2 +
6 b^6 c^2 u^2 v^2 w^2 - 6 a^4 c^4 u^2 v^2 w^2 -
6 b^4 c^4 u^2 v^2 w^2 + 2 a^2 c^6 u^2 v^2 w^2 +
2 b^2 c^6 u^2 v^2 w^2 - 2 a^8 u v^3 w^2 + 6 a^6 b^2 u v^3 w^2 -
6 a^4 b^4 u v^3 w^2 + 2 a^2 b^6 u v^3 w^2 + 5 a^6 c^2 u v^3 w^2 -
6 a^4 b^2 c^2 u v^3 w^2 - 3 a^2 b^4 c^2 u v^3 w^2 -
4 a^4 c^4 u v^3 w^2 + a^2 c^6 u v^3 w^2 + a^4 b^4 u^3 w^3 -
b^8 u^3 w^3 + 2 b^6 c^2 u^3 w^3 - b^4 c^4 u^3 w^3 +
4 a^6 b^2 u^2 v w^3 - 7 a^4 b^4 u^2 v w^3 + 4 a^2 b^6 u^2 v w^3 -
b^8 u^2 v w^3 - 7 a^4 b^2 c^2 u^2 v w^3 -
6 a^2 b^4 c^2 u^2 v w^3 + 3 b^6 c^2 u^2 v w^3 +
2 a^2 b^2 c^4 u^2 v w^3 - 3 b^4 c^4 u^2 v w^3 +
b^2 c^6 u^2 v w^3 - a^8 u v^2 w^3 + 4 a^6 b^2 u v^2 w^3 -
7 a^4 b^4 u v^2 w^3 + 4 a^2 b^6 u v^2 w^3 + 3 a^6 c^2 u v^2 w^3 -
6 a^4 b^2 c^2 u v^2 w^3 - 7 a^2 b^4 c^2 u v^2 w^3 -
3 a^4 c^4 u v^2 w^3 + 2 a^2 b^2 c^4 u v^2 w^3 +
a^2 c^6 u v^2 w^3 - a^8 v^3 w^3 + a^4 b^4 v^3 w^3 +
2 a^6 c^2 v^3 w^3 - a^4 c^4 v^3 w^3)}

For A'B'C' = circuncevian triangle of P=(u:v:w) with respect to ABC the intersection of the three radical axes is the point:

{-a^4 b^2 c^4 u^4 v^2 + 4 a^2 b^4 c^4 u^4 v^2 - 3 b^6 c^4 u^4 v^2 +
4 a^2 b^2 c^6 u^4 v^2 + 6 b^4 c^6 u^4 v^2 - 3 b^2 c^8 u^4 v^2 -
a^6 c^4 u^3 v^3 + 4 a^4 b^2 c^4 u^3 v^3 - a^2 b^4 c^4 u^3 v^3 -
2 b^6 c^4 u^3 v^3 + 4 a^4 c^6 u^3 v^3 + 6 a^2 b^2 c^6 u^3 v^3 +
6 b^4 c^6 u^3 v^3 - 5 a^2 c^8 u^3 v^3 - 6 b^2 c^8 u^3 v^3 +
2 c^10 u^3 v^3 + 2 a^4 b^2 c^4 u^2 v^4 - 2 a^2 b^4 c^4 u^2 v^4 +
2 a^4 c^6 u^2 v^4 + 4 a^2 b^2 c^6 u^2 v^4 - 2 a^2 c^8 u^2 v^4 +
2 a^6 b^2 c^2 u^4 v w - 7 a^4 b^4 c^2 u^4 v w +
8 a^2 b^6 c^2 u^4 v w - 3 b^8 c^2 u^4 v w - 7 a^4 b^2 c^4 u^4 v w +
3 b^6 c^4 u^4 v w + 8 a^2 b^2 c^6 u^4 v w + 3 b^4 c^6 u^4 v w -
3 b^2 c^8 u^4 v w + 2 a^8 c^2 u^3 v^2 w - 7 a^6 b^2 c^2 u^3 v^2 w +
8 a^4 b^4 c^2 u^3 v^2 w - 3 a^2 b^6 c^2 u^3 v^2 w -
8 a^6 c^4 u^3 v^2 w + 4 a^4 b^2 c^4 u^3 v^2 w +
2 a^2 b^4 c^4 u^3 v^2 w - 2 b^6 c^4 u^3 v^2 w +
12 a^4 c^6 u^3 v^2 w + 9 a^2 b^2 c^6 u^3 v^2 w +
6 b^4 c^6 u^3 v^2 w - 8 a^2 c^8 u^3 v^2 w - 6 b^2 c^8 u^3 v^2 w +
2 c^10 u^3 v^2 w + a^4 b^4 c^2 u^2 v^3 w - 2 a^2 b^6 c^2 u^2 v^3 w +
b^8 c^2 u^2 v^3 w + 6 a^4 b^2 c^4 u^2 v^3 w +
2 a^2 b^4 c^4 u^2 v^3 w - 4 b^6 c^4 u^2 v^3 w + a^4 c^6 u^2 v^3 w +
2 a^2 b^2 c^6 u^2 v^3 w + 6 b^4 c^6 u^2 v^3 w -
2 a^2 c^8 u^2 v^3 w - 4 b^2 c^8 u^2 v^3 w + c^10 u^2 v^3 w +
a^6 b^2 c^2 u v^4 w - 2 a^4 b^4 c^2 u v^4 w + a^2 b^6 c^2 u v^4 w +
3 a^6 c^4 u v^4 w + 4 a^4 b^2 c^4 u v^4 w - 3 a^2 b^4 c^4 u v^4 w -
2 a^4 c^6 u v^4 w + 3 a^2 b^2 c^6 u v^4 w - a^2 c^8 u v^4 w -
a^4 b^4 c^2 u^4 w^2 + 4 a^2 b^6 c^2 u^4 w^2 - 3 b^8 c^2 u^4 w^2 +
4 a^2 b^4 c^4 u^4 w^2 + 6 b^6 c^4 u^4 w^2 - 3 b^4 c^6 u^4 w^2 +
2 a^8 b^2 u^3 v w^2 - 8 a^6 b^4 u^3 v w^2 + 12 a^4 b^6 u^3 v w^2 -
8 a^2 b^8 u^3 v w^2 + 2 b^10 u^3 v w^2 - 7 a^6 b^2 c^2 u^3 v w^2 +
4 a^4 b^4 c^2 u^3 v w^2 + 9 a^2 b^6 c^2 u^3 v w^2 -
6 b^8 c^2 u^3 v w^2 + 8 a^4 b^2 c^4 u^3 v w^2 +
2 a^2 b^4 c^4 u^3 v w^2 + 6 b^6 c^4 u^3 v w^2 -
3 a^2 b^2 c^6 u^3 v w^2 - 2 b^4 c^6 u^3 v w^2 +
2 a^10 u^2 v^2 w^2 - 7 a^8 b^2 u^2 v^2 w^2 +
8 a^6 b^4 u^2 v^2 w^2 - 2 a^4 b^6 u^2 v^2 w^2 -
2 a^2 b^8 u^2 v^2 w^2 + b^10 u^2 v^2 w^2 - 7 a^8 c^2 u^2 v^2 w^2 +
6 a^4 b^4 c^2 u^2 v^2 w^2 + 4 a^2 b^6 c^2 u^2 v^2 w^2 -
3 b^8 c^2 u^2 v^2 w^2 + 8 a^6 c^4 u^2 v^2 w^2 +
6 a^4 b^2 c^4 u^2 v^2 w^2 - 4 a^2 b^4 c^4 u^2 v^2 w^2 +
2 b^6 c^4 u^2 v^2 w^2 - 2 a^4 c^6 u^2 v^2 w^2 +
4 a^2 b^2 c^6 u^2 v^2 w^2 + 2 b^4 c^6 u^2 v^2 w^2 -
2 a^2 c^8 u^2 v^2 w^2 - 3 b^2 c^8 u^2 v^2 w^2 + c^10 u^2 v^2 w^2 +
a^10 u v^3 w^2 - 4 a^8 b^2 u v^3 w^2 + 6 a^6 b^4 u v^3 w^2 -
4 a^4 b^6 u v^3 w^2 + a^2 b^8 u v^3 w^2 - 4 a^8 c^2 u v^3 w^2 +
a^6 b^2 c^2 u v^3 w^2 + 6 a^4 b^4 c^2 u v^3 w^2 -
3 a^2 b^6 c^2 u v^3 w^2 + 5 a^6 c^4 u v^3 w^2 +
3 a^2 b^4 c^4 u v^3 w^2 - 2 a^4 c^6 u v^3 w^2 -
a^2 b^2 c^6 u v^3 w^2 + a^8 c^2 v^4 w^2 - a^4 b^4 c^2 v^4 w^2 +
2 a^4 b^2 c^4 v^4 w^2 - a^4 c^6 v^4 w^2 - a^6 b^4 u^3 w^3 +
4 a^4 b^6 u^3 w^3 - 5 a^2 b^8 u^3 w^3 + 2 b^10 u^3 w^3 +
4 a^4 b^4 c^2 u^3 w^3 + 6 a^2 b^6 c^2 u^3 w^3 - 6 b^8 c^2 u^3 w^3 -
a^2 b^4 c^4 u^3 w^3 + 6 b^6 c^4 u^3 w^3 - 2 b^4 c^6 u^3 w^3 +
a^4 b^6 u^2 v w^3 - 2 a^2 b^8 u^2 v w^3 + b^10 u^2 v w^3 +
6 a^4 b^4 c^2 u^2 v w^3 + 2 a^2 b^6 c^2 u^2 v w^3 -
4 b^8 c^2 u^2 v w^3 + a^4 b^2 c^4 u^2 v w^3 +
2 a^2 b^4 c^4 u^2 v w^3 + 6 b^6 c^4 u^2 v w^3 -
2 a^2 b^2 c^6 u^2 v w^3 - 4 b^4 c^6 u^2 v w^3 + b^2 c^8 u^2 v w^3 +
a^10 u v^2 w^3 - 4 a^8 b^2 u v^2 w^3 + 5 a^6 b^4 u v^2 w^3 -
2 a^4 b^6 u v^2 w^3 - 4 a^8 c^2 u v^2 w^3 + a^6 b^2 c^2 u v^2 w^3 -
a^2 b^6 c^2 u v^2 w^3 + 6 a^6 c^4 u v^2 w^3 +
6 a^4 b^2 c^4 u v^2 w^3 + 3 a^2 b^4 c^4 u v^2 w^3 -
4 a^4 c^6 u v^2 w^3 - 3 a^2 b^2 c^6 u v^2 w^3 + a^2 c^8 u v^2 w^3 -
a^8 b^2 v^3 w^3 + 2 a^6 b^4 v^3 w^3 - a^4 b^6 v^3 w^3 -
a^8 c^2 v^3 w^3 - 4 a^6 b^2 c^2 v^3 w^3 + a^4 b^4 c^2 v^3 w^3 +
2 a^6 c^4 v^3 w^3 + a^4 b^2 c^4 v^3 w^3 - a^4 c^6 v^3 w^3 +
2 a^4 b^6 u^2 w^4 - 2 a^2 b^8 u^2 w^4 + 2 a^4 b^4 c^2 u^2 w^4 +
4 a^2 b^6 c^2 u^2 w^4 - 2 a^2 b^4 c^4 u^2 w^4 + 3 a^6 b^4 u v w^4 -
2 a^4 b^6 u v w^4 - a^2 b^8 u v w^4 + a^6 b^2 c^2 u v w^4 +
4 a^4 b^4 c^2 u v w^4 + 3 a^2 b^6 c^2 u v w^4 -
2 a^4 b^2 c^4 u v w^4 - 3 a^2 b^4 c^4 u v w^4 +
a^2 b^2 c^6 u v w^4 + a^8 b^2 v^2 w^4 - a^4 b^6 v^2 w^4 +
2 a^4 b^4 c^2 v^2 w^4 -
a^4 b^2 c^4 v^2 w^4, -2 a^4 b^2 c^4 u^4 v^2 +
2 a^2 b^4 c^4 u^4 v^2 + 4 a^2 b^2 c^6 u^4 v^2 + 2 b^4 c^6 u^4 v^2 -
2 b^2 c^8 u^4 v^2 - 2 a^6 c^4 u^3 v^3 - a^4 b^2 c^4 u^3 v^3 +
4 a^2 b^4 c^4 u^3 v^3 - b^6 c^4 u^3 v^3 + 6 a^4 c^6 u^3 v^3 +
6 a^2 b^2 c^6 u^3 v^3 + 4 b^4 c^6 u^3 v^3 - 6 a^2 c^8 u^3 v^3 -
5 b^2 c^8 u^3 v^3 + 2 c^10 u^3 v^3 - 3 a^6 c^4 u^2 v^4 +
4 a^4 b^2 c^4 u^2 v^4 - a^2 b^4 c^4 u^2 v^4 + 6 a^4 c^6 u^2 v^4 +
4 a^2 b^2 c^6 u^2 v^4 - 3 a^2 c^8 u^2 v^4 + a^6 b^2 c^2 u^4 v w -
2 a^4 b^4 c^2 u^4 v w + a^2 b^6 c^2 u^4 v w -
3 a^4 b^2 c^4 u^4 v w + 4 a^2 b^4 c^4 u^4 v w + 3 b^6 c^4 u^4 v w +
3 a^2 b^2 c^6 u^4 v w - 2 b^4 c^6 u^4 v w - b^2 c^8 u^4 v w +
a^8 c^2 u^3 v^2 w - 2 a^6 b^2 c^2 u^3 v^2 w +
a^4 b^4 c^2 u^3 v^2 w - 4 a^6 c^4 u^3 v^2 w +
2 a^4 b^2 c^4 u^3 v^2 w + 6 a^2 b^4 c^4 u^3 v^2 w +
6 a^4 c^6 u^3 v^2 w + 2 a^2 b^2 c^6 u^3 v^2 w + b^4 c^6 u^3 v^2 w -
4 a^2 c^8 u^3 v^2 w - 2 b^2 c^8 u^3 v^2 w + c^10 u^3 v^2 w -
3 a^6 b^2 c^2 u^2 v^3 w + 8 a^4 b^4 c^2 u^2 v^3 w -
7 a^2 b^6 c^2 u^2 v^3 w + 2 b^8 c^2 u^2 v^3 w -
2 a^6 c^4 u^2 v^3 w + 2 a^4 b^2 c^4 u^2 v^3 w +
4 a^2 b^4 c^4 u^2 v^3 w - 8 b^6 c^4 u^2 v^3 w +
6 a^4 c^6 u^2 v^3 w + 9 a^2 b^2 c^6 u^2 v^3 w +
12 b^4 c^6 u^2 v^3 w - 6 a^2 c^8 u^2 v^3 w - 8 b^2 c^8 u^2 v^3 w +
2 c^10 u^2 v^3 w - 3 a^8 c^2 u v^4 w + 8 a^6 b^2 c^2 u v^4 w -
7 a^4 b^4 c^2 u v^4 w + 2 a^2 b^6 c^2 u v^4 w + 3 a^6 c^4 u v^4 w -
7 a^2 b^4 c^4 u v^4 w + 3 a^4 c^6 u v^4 w + 8 a^2 b^2 c^6 u v^4 w -
3 a^2 c^8 u v^4 w - a^4 b^4 c^2 u^4 w^2 + b^8 c^2 u^4 w^2 +
2 a^2 b^4 c^4 u^4 w^2 - b^4 c^6 u^4 w^2 + a^8 b^2 u^3 v w^2 -
4 a^6 b^4 u^3 v w^2 + 6 a^4 b^6 u^3 v w^2 - 4 a^2 b^8 u^3 v w^2 +
b^10 u^3 v w^2 - 3 a^6 b^2 c^2 u^3 v w^2 + 6 a^4 b^4 c^2 u^3 v w^2 +
a^2 b^6 c^2 u^3 v w^2 - 4 b^8 c^2 u^3 v w^2 +
3 a^4 b^2 c^4 u^3 v w^2 + 5 b^6 c^4 u^3 v w^2 -
a^2 b^2 c^6 u^3 v w^2 - 2 b^4 c^6 u^3 v w^2 + a^10 u^2 v^2 w^2 -
2 a^8 b^2 u^2 v^2 w^2 - 2 a^6 b^4 u^2 v^2 w^2 +
8 a^4 b^6 u^2 v^2 w^2 - 7 a^2 b^8 u^2 v^2 w^2 +
2 b^10 u^2 v^2 w^2 - 3 a^8 c^2 u^2 v^2 w^2 +
4 a^6 b^2 c^2 u^2 v^2 w^2 + 6 a^4 b^4 c^2 u^2 v^2 w^2 -
7 b^8 c^2 u^2 v^2 w^2 + 2 a^6 c^4 u^2 v^2 w^2 -
4 a^4 b^2 c^4 u^2 v^2 w^2 + 6 a^2 b^4 c^4 u^2 v^2 w^2 +
8 b^6 c^4 u^2 v^2 w^2 + 2 a^4 c^6 u^2 v^2 w^2 +
4 a^2 b^2 c^6 u^2 v^2 w^2 - 2 b^4 c^6 u^2 v^2 w^2 -
3 a^2 c^8 u^2 v^2 w^2 - 2 b^2 c^8 u^2 v^2 w^2 + c^10 u^2 v^2 w^2 +
2 a^10 u v^3 w^2 - 8 a^8 b^2 u v^3 w^2 + 12 a^6 b^4 u v^3 w^2 -
8 a^4 b^6 u v^3 w^2 + 2 a^2 b^8 u v^3 w^2 - 6 a^8 c^2 u v^3 w^2 +
9 a^6 b^2 c^2 u v^3 w^2 + 4 a^4 b^4 c^2 u v^3 w^2 -
7 a^2 b^6 c^2 u v^3 w^2 + 6 a^6 c^4 u v^3 w^2 +
2 a^4 b^2 c^4 u v^3 w^2 + 8 a^2 b^4 c^4 u v^3 w^2 -
2 a^4 c^6 u v^3 w^2 - 3 a^2 b^2 c^6 u v^3 w^2 - 3 a^8 c^2 v^4 w^2 +
4 a^6 b^2 c^2 v^4 w^2 - a^4 b^4 c^2 v^4 w^2 + 6 a^6 c^4 v^4 w^2 +
4 a^4 b^2 c^4 v^4 w^2 - 3 a^4 c^6 v^4 w^2 - a^6 b^4 u^3 w^3 +
2 a^4 b^6 u^3 w^3 - a^2 b^8 u^3 w^3 + a^4 b^4 c^2 u^3 w^3 -
4 a^2 b^6 c^2 u^3 w^3 - b^8 c^2 u^3 w^3 + a^2 b^4 c^4 u^3 w^3 +
2 b^6 c^4 u^3 w^3 - b^4 c^6 u^3 w^3 - 2 a^6 b^4 u^2 v w^3 +
5 a^4 b^6 u^2 v w^3 - 4 a^2 b^8 u^2 v w^3 + b^10 u^2 v w^3 -
a^6 b^2 c^2 u^2 v w^3 + a^2 b^6 c^2 u^2 v w^3 -
4 b^8 c^2 u^2 v w^3 + 3 a^4 b^2 c^4 u^2 v w^3 +
6 a^2 b^4 c^4 u^2 v w^3 + 6 b^6 c^4 u^2 v w^3 -
3 a^2 b^2 c^6 u^2 v w^3 - 4 b^4 c^6 u^2 v w^3 + b^2 c^8 u^2 v w^3 +
a^10 u v^2 w^3 - 2 a^8 b^2 u v^2 w^3 + a^6 b^4 u v^2 w^3 -
4 a^8 c^2 u v^2 w^3 + 2 a^6 b^2 c^2 u v^2 w^3 +
6 a^4 b^4 c^2 u v^2 w^3 + 6 a^6 c^4 u v^2 w^3 +
2 a^4 b^2 c^4 u v^2 w^3 + a^2 b^4 c^4 u v^2 w^3 -
4 a^4 c^6 u v^2 w^3 - 2 a^2 b^2 c^6 u v^2 w^3 + a^2 c^8 u v^2 w^3 +
2 a^10 v^3 w^3 - 5 a^8 b^2 v^3 w^3 + 4 a^6 b^4 v^3 w^3 -
a^4 b^6 v^3 w^3 - 6 a^8 c^2 v^3 w^3 + 6 a^6 b^2 c^2 v^3 w^3 +
4 a^4 b^4 c^2 v^3 w^3 + 6 a^6 c^4 v^3 w^3 - a^4 b^2 c^4 v^3 w^3 -
2 a^4 c^6 v^3 w^3 - a^6 b^4 u^2 w^4 + a^2 b^8 u^2 w^4 +
2 a^4 b^4 c^2 u^2 w^4 - a^2 b^4 c^4 u^2 w^4 - a^8 b^2 u v w^4 -
2 a^6 b^4 u v w^4 + 3 a^4 b^6 u v w^4 + 3 a^6 b^2 c^2 u v w^4 +
4 a^4 b^4 c^2 u v w^4 + a^2 b^6 c^2 u v w^4 -
3 a^4 b^2 c^4 u v w^4 - 2 a^2 b^4 c^4 u v w^4 +
a^2 b^2 c^6 u v w^4 - 2 a^8 b^2 v^2 w^4 + 2 a^6 b^4 v^2 w^4 +
4 a^6 b^2 c^2 v^2 w^4 + 2 a^4 b^4 c^2 v^2 w^4 -
2 a^4 b^2 c^4 v^2 w^4, -a^4 b^2 c^4 u^4 v^2 +
2 a^2 b^4 c^4 u^4 v^2 - b^6 c^4 u^4 v^2 + b^2 c^8 u^4 v^2 -
a^6 c^4 u^3 v^3 + a^4 b^2 c^4 u^3 v^3 + a^2 b^4 c^4 u^3 v^3 -
b^6 c^4 u^3 v^3 + 2 a^4 c^6 u^3 v^3 - 4 a^2 b^2 c^6 u^3 v^3 +
2 b^4 c^6 u^3 v^3 - a^2 c^8 u^3 v^3 - b^2 c^8 u^3 v^3 -
a^6 c^4 u^2 v^4 + 2 a^4 b^2 c^4 u^2 v^4 - a^2 b^4 c^4 u^2 v^4 +
a^2 c^8 u^2 v^4 + a^6 b^2 c^2 u^4 v w - 3 a^4 b^4 c^2 u^4 v w +
3 a^2 b^6 c^2 u^4 v w - b^8 c^2 u^4 v w - 2 a^4 b^2 c^4 u^4 v w +
4 a^2 b^4 c^4 u^4 v w - 2 b^6 c^4 u^4 v w + a^2 b^2 c^6 u^4 v w +
3 b^4 c^6 u^4 v w + a^8 c^2 u^3 v^2 w - 3 a^6 b^2 c^2 u^3 v^2 w +
3 a^4 b^4 c^2 u^3 v^2 w - a^2 b^6 c^2 u^3 v^2 w -
4 a^6 c^4 u^3 v^2 w + 6 a^4 b^2 c^4 u^3 v^2 w -
2 b^6 c^4 u^3 v^2 w + 6 a^4 c^6 u^3 v^2 w + a^2 b^2 c^6 u^3 v^2 w +
5 b^4 c^6 u^3 v^2 w - 4 a^2 c^8 u^3 v^2 w - 4 b^2 c^8 u^3 v^2 w +
c^10 u^3 v^2 w - a^6 b^2 c^2 u^2 v^3 w + 3 a^4 b^4 c^2 u^2 v^3 w -
3 a^2 b^6 c^2 u^2 v^3 w + b^8 c^2 u^2 v^3 w - 2 a^6 c^4 u^2 v^3 w +
6 a^2 b^4 c^4 u^2 v^3 w - 4 b^6 c^4 u^2 v^3 w +
5 a^4 c^6 u^2 v^3 w + a^2 b^2 c^6 u^2 v^3 w + 6 b^4 c^6 u^2 v^3 w -
4 a^2 c^8 u^2 v^3 w - 4 b^2 c^8 u^2 v^3 w + c^10 u^2 v^3 w -
a^8 c^2 u v^4 w + 3 a^6 b^2 c^2 u v^4 w - 3 a^4 b^4 c^2 u v^4 w +
a^2 b^6 c^2 u v^4 w - 2 a^6 c^4 u v^4 w + 4 a^4 b^2 c^4 u v^4 w -
2 a^2 b^4 c^4 u v^4 w + 3 a^4 c^6 u v^4 w + a^2 b^2 c^6 u v^4 w -
2 a^4 b^4 c^2 u^4 w^2 + 4 a^2 b^6 c^2 u^4 w^2 - 2 b^8 c^2 u^4 w^2 +
2 a^2 b^4 c^4 u^4 w^2 + 2 b^6 c^4 u^4 w^2 + a^8 b^2 u^3 v w^2 -
4 a^6 b^4 u^3 v w^2 + 6 a^4 b^6 u^3 v w^2 - 4 a^2 b^8 u^3 v w^2 +
b^10 u^3 v w^2 - 2 a^6 b^2 c^2 u^3 v w^2 +
2 a^4 b^4 c^2 u^3 v w^2 + 2 a^2 b^6 c^2 u^3 v w^2 -
2 b^8 c^2 u^3 v w^2 + a^4 b^2 c^4 u^3 v w^2 +
6 a^2 b^4 c^4 u^3 v w^2 + b^6 c^4 u^3 v w^2 + a^10 u^2 v^2 w^2 -
3 a^8 b^2 u^2 v^2 w^2 + 2 a^6 b^4 u^2 v^2 w^2 +
2 a^4 b^6 u^2 v^2 w^2 - 3 a^2 b^8 u^2 v^2 w^2 + b^10 u^2 v^2 w^2 -
2 a^8 c^2 u^2 v^2 w^2 + 4 a^6 b^2 c^2 u^2 v^2 w^2 -
4 a^4 b^4 c^2 u^2 v^2 w^2 + 4 a^2 b^6 c^2 u^2 v^2 w^2 -
2 b^8 c^2 u^2 v^2 w^2 - 2 a^6 c^4 u^2 v^2 w^2 +
6 a^4 b^2 c^4 u^2 v^2 w^2 + 6 a^2 b^4 c^4 u^2 v^2 w^2 -
2 b^6 c^4 u^2 v^2 w^2 + 8 a^4 c^6 u^2 v^2 w^2 +
8 b^4 c^6 u^2 v^2 w^2 - 7 a^2 c^8 u^2 v^2 w^2 -
7 b^2 c^8 u^2 v^2 w^2 + 2 c^10 u^2 v^2 w^2 + a^10 u v^3 w^2 -
4 a^8 b^2 u v^3 w^2 + 6 a^6 b^4 u v^3 w^2 - 4 a^4 b^6 u v^3 w^2 +
a^2 b^8 u v^3 w^2 - 2 a^8 c^2 u v^3 w^2 + 2 a^6 b^2 c^2 u v^3 w^2 +
2 a^4 b^4 c^2 u v^3 w^2 - 2 a^2 b^6 c^2 u v^3 w^2 +
a^6 c^4 u v^3 w^2 + 6 a^4 b^2 c^4 u v^3 w^2 +
a^2 b^4 c^4 u v^3 w^2 - 2 a^8 c^2 v^4 w^2 + 4 a^6 b^2 c^2 v^4 w^2 -
2 a^4 b^4 c^2 v^4 w^2 + 2 a^6 c^4 v^4 w^2 + 2 a^4 b^2 c^4 v^4 w^2 -
2 a^6 b^4 u^3 w^3 + 6 a^4 b^6 u^3 w^3 - 6 a^2 b^8 u^3 w^3 +
2 b^10 u^3 w^3 - a^4 b^4 c^2 u^3 w^3 + 6 a^2 b^6 c^2 u^3 w^3 -
5 b^8 c^2 u^3 w^3 + 4 a^2 b^4 c^4 u^3 w^3 + 4 b^6 c^4 u^3 w^3 -
b^4 c^6 u^3 w^3 - 2 a^6 b^4 u^2 v w^3 + 6 a^4 b^6 u^2 v w^3 -
6 a^2 b^8 u^2 v w^3 + 2 b^10 u^2 v w^3 - 3 a^6 b^2 c^2 u^2 v w^3 +
2 a^4 b^4 c^2 u^2 v w^3 + 9 a^2 b^6 c^2 u^2 v w^3 -
8 b^8 c^2 u^2 v w^3 + 8 a^4 b^2 c^4 u^2 v w^3 +
4 a^2 b^4 c^4 u^2 v w^3 + 12 b^6 c^4 u^2 v w^3 -
7 a^2 b^2 c^6 u^2 v w^3 - 8 b^4 c^6 u^2 v w^3 +
2 b^2 c^8 u^2 v w^3 + 2 a^10 u v^2 w^3 - 6 a^8 b^2 u v^2 w^3 +
6 a^6 b^4 u v^2 w^3 - 2 a^4 b^6 u v^2 w^3 - 8 a^8 c^2 u v^2 w^3 +
9 a^6 b^2 c^2 u v^2 w^3 + 2 a^4 b^4 c^2 u v^2 w^3 -
3 a^2 b^6 c^2 u v^2 w^3 + 12 a^6 c^4 u v^2 w^3 +
4 a^4 b^2 c^4 u v^2 w^3 + 8 a^2 b^4 c^4 u v^2 w^3 -
8 a^4 c^6 u v^2 w^3 - 7 a^2 b^2 c^6 u v^2 w^3 +
2 a^2 c^8 u v^2 w^3 + 2 a^10 v^3 w^3 - 6 a^8 b^2 v^3 w^3 +
6 a^6 b^4 v^3 w^3 - 2 a^4 b^6 v^3 w^3 - 5 a^8 c^2 v^3 w^3 +
6 a^6 b^2 c^2 v^3 w^3 - a^4 b^4 c^2 v^3 w^3 + 4 a^6 c^4 v^3 w^3 +
4 a^4 b^2 c^4 v^3 w^3 - a^4 c^6 v^3 w^3 - 3 a^6 b^4 u^2 w^4 +
6 a^4 b^6 u^2 w^4 - 3 a^2 b^8 u^2 w^4 + 4 a^4 b^4 c^2 u^2 w^4 +
4 a^2 b^6 c^2 u^2 w^4 - a^2 b^4 c^4 u^2 w^4 - 3 a^8 b^2 u v w^4 +
3 a^6 b^4 u v w^4 + 3 a^4 b^6 u v w^4 - 3 a^2 b^8 u v w^4 +
8 a^6 b^2 c^2 u v w^4 + 8 a^2 b^6 c^2 u v w^4 -
7 a^4 b^2 c^4 u v w^4 - 7 a^2 b^4 c^4 u v w^4 +
2 a^2 b^2 c^6 u v w^4 - 3 a^8 b^2 v^2 w^4 + 6 a^6 b^4 v^2 w^4 -
3 a^4 b^6 v^2 w^4 + 4 a^6 b^2 c^2 v^2 w^4 + 4 a^4 b^4 c^2 v^2 w^4 -
a^4 b^2 c^4 v^2 w^4}

Francisco Javier García Capitán
15 December 2011

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Generalization:

Let 123456 be a cyclic hexagon (ie inscribed in a circle).

The radical axes of the NPCs of the pairs of the triangles:

abc, def, where {a,b,c,d,e,f} = {1,2,3,4,5,6} (ie pairs of triangles with no common vertex) are concurrent (at the common midpoint of the distances of the centers of the pairs of the NPCs).

We have 10 pairs of triangles:

(123,456), (124,356), (125,346), (126,345)

(134,256), (135,246), (136,245)

(145,236), (146,235)

(156,234)

The centers of the 10 NPCs lie on a conic centered at the point of concurrence of the radical axes.

APH, 17 December 2011

Reflctions of AO,BO,CO in BC,CA,AB

Let ABC be a triangle, La the reflection of AO in BC, Lab,Lac the reflections of La in BO, CO, resp. and A' := Lab /\ Lac. Similarly B',C'.

The triangles ABC, A'B'C' are perspective.

Perspector ?

APH, 13 December 2011

--------------------------------------------

The perspector is X26.

The locus is a 16th curve through O and H, which is a trivial case.
Francisco Javier García Capitán
14 December 2011

Parallel Lines : GENERALIZATION 2

[APH]:
> > Let ABC be a triangle and L a line.

> >
> > L1,L2,L3 := the reflections of AI, BI, CI in L, resp.
> >
> > M1,M2,M3 := the reflections of AH, BH, CH in L1, L2, L3, resp
then M1,M2,M3 are parallel.

[Jean-Pierre Ehrmann]:
If (L,L') is the directed angle (mod Pi) between the lines L & L', we have
(M1,M2)+(HA,HB)=2(L1,L2)=2(BI,AI)=(CB,CA)=(HA,HB) thus (M1,M2)=0

Something curious : it seems that, if we take X(80) (reflection of the incenter in the Feuerbach point) instead of H, the lines M1,M2,M3 concur for every line L. But why?

Hyacinthos #20524

[Francisco Javier]:

[APH]:
>It seems that H and X(80) are points of the locus:
>Let ABC be a triangle P a variable point and L a fixed line.
>Let L1, L2, L3 be the reflections of AI,BI,CI in L and
>M1, M2, M3 the reflections of AP,BP,CP in L1,L2,L3, resp.
>Which is the locus of P such that M1,M2,M3 are concurrent?

For each line L: u x + v y + w z = 0 the locus of P=(x:y:z) is a cubic (see
below).

Both X4 and X80 lie on any of these cubics. From a quick sketch, these are the only points lying on all cubics.

Equation of cubic for L: u x + v y + w z = 0 is as follows:

a^5 u^2 x^2 y - 2 a^3 b^2 u^2 x^2 y + a b^4 u^2 x^2 y +
a^3 b c u^2 x^2 y - a b^3 c u^2 x^2 y + a b c^3 u^2 x^2 y -
a c^4 u^2 x^2 y - a^5 u v x^2 y + a^4 b u v x^2 y +
2 a^3 b^2 u v x^2 y - 2 a^2 b^3 u v x^2 y - a b^4 u v x^2 y +
b^5 u v x^2 y - 2 a^3 b c u v x^2 y + 2 a b^3 c u v x^2 y +
2 a^3 c^2 u v x^2 y + a^2 b c^2 u v x^2 y - 2 a b^2 c^2 u v x^2 y -
b^3 c^2 u v x^2 y - a c^4 u v x^2 y - a^4 b v^2 x^2 y +
2 a^2 b^3 v^2 x^2 y - b^5 v^2 x^2 y + a^3 b c v^2 x^2 y -
a b^3 c v^2 x^2 y + a^2 b c^2 v^2 x^2 y - b^3 c^2 v^2 x^2 y -
a b c^3 v^2 x^2 y - 2 a^3 c^2 u w x^2 y + 2 a b^2 c^2 u w x^2 y +
a^2 c^3 u w x^2 y - 2 a b c^3 u w x^2 y + b^2 c^3 u w x^2 y +
2 a c^4 u w x^2 y - c^5 u w x^2 y - 2 a^2 b c^2 v w x^2 y +
2 b^3 c^2 v w x^2 y + 2 a b c^3 v w x^2 y - 2 b^2 c^3 v w x^2 y -
a^2 c^3 w^2 x^2 y + b^2 c^3 w^2 x^2 y + c^5 w^2 x^2 y +
a^5 u^2 x y^2 - 2 a^3 b^2 u^2 x y^2 + a b^4 u^2 x y^2 +
a^3 b c u^2 x y^2 - a b^3 c u^2 x y^2 + a^3 c^2 u^2 x y^2 -
a b^2 c^2 u^2 x y^2 + a b c^3 u^2 x y^2 - a^5 u v x y^2 +
a^4 b u v x y^2 + 2 a^3 b^2 u v x y^2 - 2 a^2 b^3 u v x y^2 -
a b^4 u v x y^2 + b^5 u v x y^2 - 2 a^3 b c u v x y^2 +
2 a b^3 c u v x y^2 + a^3 c^2 u v x y^2 + 2 a^2 b c^2 u v x y^2 -
a b^2 c^2 u v x y^2 - 2 b^3 c^2 u v x y^2 + b c^4 u v x y^2 -
a^4 b v^2 x y^2 + 2 a^2 b^3 v^2 x y^2 - b^5 v^2 x y^2 +
a^3 b c v^2 x y^2 - a b^3 c v^2 x y^2 - a b c^3 v^2 x y^2 +
b c^4 v^2 x y^2 - 2 a^3 c^2 u w x y^2 + 2 a b^2 c^2 u w x y^2 +
2 a^2 c^3 u w x y^2 - 2 a b c^3 u w x y^2 - 2 a^2 b c^2 v w x y^2 +
2 b^3 c^2 v w x y^2 - a^2 c^3 v w x y^2 + 2 a b c^3 v w x y^2 -
b^2 c^3 v w x y^2 - 2 b c^4 v w x y^2 + c^5 v w x y^2 -
a^2 c^3 w^2 x y^2 + b^2 c^3 w^2 x y^2 - c^5 w^2 x y^2 -
a^5 u^2 x^2 z + a b^4 u^2 x^2 z - a^3 b c u^2 x^2 z -
a b^3 c u^2 x^2 z + 2 a^3 c^2 u^2 x^2 z + a b c^3 u^2 x^2 z -
a c^4 u^2 x^2 z + 2 a^3 b^2 u v x^2 z - a^2 b^3 u v x^2 z -
2 a b^4 u v x^2 z + b^5 u v x^2 z + 2 a b^3 c u v x^2 z -
2 a b^2 c^2 u v x^2 z - b^3 c^2 u v x^2 z + a^2 b^3 v^2 x^2 z -
b^5 v^2 x^2 z - b^3 c^2 v^2 x^2 z + a^5 u w x^2 z -
2 a^3 b^2 u w x^2 z + a b^4 u w x^2 z - a^4 c u w x^2 z +
2 a^3 b c u w x^2 z - a^2 b^2 c u w x^2 z - 2 a^3 c^2 u w x^2 z +
2 a b^2 c^2 u w x^2 z + 2 a^2 c^3 u w x^2 z - 2 a b c^3 u w x^2 z +
b^2 c^3 u w x^2 z + a c^4 u w x^2 z - c^5 u w x^2 z +
2 a^2 b^2 c v w x^2 z - 2 a b^3 c v w x^2 z + 2 b^3 c^2 v w x^2 z -
2 b^2 c^3 v w x^2 z + a^4 c w^2 x^2 z - a^3 b c w^2 x^2 z -
a^2 b^2 c w^2 x^2 z + a b^3 c w^2 x^2 z - 2 a^2 c^3 w^2 x^2 z +
a b c^3 w^2 x^2 z + b^2 c^3 w^2 x^2 z + c^5 w^2 x^2 z -
a^3 b^2 u^2 x y z + a b^4 u^2 x y z - 2 a b^3 c u^2 x y z +
a^3 c^2 u^2 x y z + 2 a b c^3 u^2 x y z - a c^4 u^2 x y z -
a^5 u v x y z + 2 a^4 b u v x y z + 3 a^3 b^2 u v x y z -
3 a^2 b^3 u v x y z - 2 a b^4 u v x y z + b^5 u v x y z -
4 a^3 b c u v x y z + 4 a b^3 c u v x y z + 2 a^3 c^2 u v x y z +
a^2 b c^2 u v x y z - a b^2 c^2 u v x y z - 2 b^3 c^2 u v x y z -
a c^4 u v x y z + b c^4 u v x y z - a^4 b v^2 x y z +
a^2 b^3 v^2 x y z + 2 a^3 b c v^2 x y z - b^3 c^2 v^2 x y z -
2 a b c^3 v^2 x y z + b c^4 v^2 x y z + a^5 u w x y z -
2 a^3 b^2 u w x y z + a b^4 u w x y z - 2 a^4 c u w x y z +
4 a^3 b c u w x y z - a^2 b^2 c u w x y z - b^4 c u w x y z -
3 a^3 c^2 u w x y z + a b^2 c^2 u w x y z + 3 a^2 c^3 u w x y z -
4 a b c^3 u w x y z + 2 b^2 c^3 u w x y z + 2 a c^4 u w x y z -
c^5 u w x y z - a^4 b v w x y z + 2 a^2 b^3 v w x y z -
b^5 v w x y z + a^4 c v w x y z + a^2 b^2 c v w x y z -
4 a b^3 c v w x y z + 2 b^4 c v w x y z - a^2 b c^2 v w x y z +
3 b^3 c^2 v w x y z - 2 a^2 c^3 v w x y z + 4 a b c^3 v w x y z -
3 b^2 c^3 v w x y z - 2 b c^4 v w x y z + c^5 v w x y z +
a^4 c w^2 x y z - 2 a^3 b c w^2 x y z + 2 a b^3 c w^2 x y z -
b^4 c w^2 x y z - a^2 c^3 w^2 x y z + b^2 c^3 w^2 x y z +
a^5 u^2 y^2 z - a^3 b^2 u^2 y^2 z + a^3 c^2 u^2 y^2 z -
a^5 u v y^2 z + 2 a^4 b u v y^2 z + a^3 b^2 u v y^2 z -
2 a^2 b^3 u v y^2 z - 2 a^3 b c u v y^2 z + a^3 c^2 u v y^2 z +
2 a^2 b c^2 u v y^2 z - a^4 b v^2 y^2 z + b^5 v^2 y^2 z +
a^3 b c v^2 y^2 z + a b^3 c v^2 y^2 z - 2 b^3 c^2 v^2 y^2 z -
a b c^3 v^2 y^2 z + b c^4 v^2 y^2 z + 2 a^3 b c u w y^2 z -
2 a^2 b^2 c u w y^2 z - 2 a^3 c^2 u w y^2 z + 2 a^2 c^3 u w y^2 z -
a^4 b v w y^2 z + 2 a^2 b^3 v w y^2 z - b^5 v w y^2 z +
a^2 b^2 c v w y^2 z - 2 a b^3 c v w y^2 z + b^4 c v w y^2 z -
2 a^2 b c^2 v w y^2 z + 2 b^3 c^2 v w y^2 z - a^2 c^3 v w y^2 z +
2 a b c^3 v w y^2 z - 2 b^2 c^3 v w y^2 z - b c^4 v w y^2 z +
c^5 v w y^2 z - a^3 b c w^2 y^2 z + a^2 b^2 c w^2 y^2 z +
a b^3 c w^2 y^2 z - b^4 c w^2 y^2 z - a^2 c^3 w^2 y^2 z -
a b c^3 w^2 y^2 z + 2 b^2 c^3 w^2 y^2 z - c^5 w^2 y^2 z -
a^5 u^2 x z^2 - a^3 b^2 u^2 x z^2 - a^3 b c u^2 x z^2 -
a b^3 c u^2 x z^2 + 2 a^3 c^2 u^2 x z^2 + a b^2 c^2 u^2 x z^2 +
a b c^3 u^2 x z^2 - a c^4 u^2 x z^2 + 2 a^3 b^2 u v x z^2 -
2 a^2 b^3 u v x z^2 + 2 a b^3 c u v x z^2 - 2 a b^2 c^2 u v x z^2 +
a^2 b^3 v^2 x z^2 + b^5 v^2 x z^2 - b^3 c^2 v^2 x z^2 +
a^5 u w x z^2 - a^3 b^2 u w x z^2 - a^4 c u w x z^2 +
2 a^3 b c u w x z^2 - 2 a^2 b^2 c u w x z^2 - b^4 c u w x z^2 -
2 a^3 c^2 u w x z^2 + a b^2 c^2 u w x z^2 + 2 a^2 c^3 u w x z^2 -
2 a b c^3 u w x z^2 + 2 b^2 c^3 u w x z^2 + a c^4 u w x z^2 -
c^5 u w x z^2 + a^2 b^3 v w x z^2 - b^5 v w x z^2 +
2 a^2 b^2 c v w x z^2 - 2 a b^3 c v w x z^2 + 2 b^4 c v w x z^2 +
b^3 c^2 v w x z^2 - 2 b^2 c^3 v w x z^2 + a^4 c w^2 x z^2 -
a^3 b c w^2 x z^2 + a b^3 c w^2 x z^2 - b^4 c w^2 x z^2 -
2 a^2 c^3 w^2 x z^2 + a b c^3 w^2 x z^2 + c^5 w^2 x z^2 -
a^5 u^2 y z^2 - a^3 b^2 u^2 y z^2 + a^3 c^2 u^2 y z^2 +
2 a^3 b^2 u v y z^2 - 2 a^2 b^3 u v y z^2 - 2 a^3 b c u v y z^2 +
2 a^2 b c^2 u v y z^2 + a^2 b^3 v^2 y z^2 + b^5 v^2 y z^2 +
a^3 b c v^2 y z^2 + a b^3 c v^2 y z^2 - a^2 b c^2 v^2 y z^2 -
2 b^3 c^2 v^2 y z^2 - a b c^3 v^2 y z^2 + b c^4 v^2 y z^2 +
a^5 u w y z^2 - a^3 b^2 u w y z^2 - 2 a^4 c u w y z^2 +
2 a^3 b c u w y z^2 - 2 a^2 b^2 c u w y z^2 - a^3 c^2 u w y z^2 +
2 a^2 c^3 u w y z^2 + a^2 b^3 v w y z^2 - b^5 v w y z^2 +
a^4 c v w y z^2 + 2 a^2 b^2 c v w y z^2 - 2 a b^3 c v w y z^2 +
b^4 c v w y z^2 - a^2 b c^2 v w y z^2 + 2 b^3 c^2 v w y z^2 -
2 a^2 c^3 v w y z^2 + 2 a b c^3 v w y z^2 - 2 b^2 c^3 v w y z^2 -
b c^4 v w y z^2 + c^5 v w y z^2 + a^4 c w^2 y z^2 -
a^3 b c w^2 y z^2 + a b^3 c w^2 y z^2 - b^4 c w^2 y z^2 -
a b c^3 w^2 y z^2 + 2 b^2 c^3 w^2 y z^2 - c^5 w^2 y z^2 = 0.

Hyacinthos #20527

Francisco Javier García Capitán
12 December 2011

[Jean-Pierre Ehrmann]:
Suppose that P & P' are antipodes upon a rectangular hyperbola.
Consider a line L and a variable point M upon the hyperbola; L'= reflection of MP wrt L; then the reflection of MQ wrt L' goes through a fixed point when M moves upon the hyperbola (this can be shown with an easy computation. Synthetic proof?). If we take successively for M the infinite points of the asymptots, we get an easy localization of the common point.

From this, it follows that if we consider a triangle ABC, a point P, its antigonal P' and a line L. If L1,L2,L3 are the reflections of AP,BP,CP wrt L, then the reflections of AP' wrt L1, of BP' wrt L2, of CP' wrt L3 are concurrent (and if, instead of ABC, we take any triangle inscribed in the rectangular circumhyperbola going through P, the common point will be the same one)
For instance, if P = I, then P'=X[80]

Hyacinthos 20538

Parallel Lines : GENERALIZATION 1

Let ABC be a triangle, P a point, A'B'C' the pedal triangle of P and L a line.

Denote (for I, J : the incenters of ABC, A'B'C', resp.):
L1,L2,L3 := the reflections of AI, BI, CI in L, resp.
M1,M2,M3 := the reflections of the bisectors A'J, B'J, C'J of the triangle A'B'C' in L1, L2, L3, resp

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

APH, 12 December 2011

Parallel Lines

Let ABC be a triangle and L a line.

Denote:
L1,L2,L3 := the reflections of AI, BI, CI in L, resp.
M1,M2,M3 := the reflections of AH, BH, CH in L1, L2, L3, resp

Point of concurrence of M1,M2,M3 ?

Special Cases: L = OH or OI or OK lines

APH, 11 December 2011

Generalization 1
Generalization 2

********************************************

I think that my calculations are right, no interesting point in any case.

OH:

{-a^14 b^2 + 5 a^12 b^4 - 10 a^10 b^6 + 10 a^8 b^8 - 5 a^6 b^10 +
a^4 b^12 - a^14 c^2 - 2 a^12 b^2 c^2 + 4 a^10 b^4 c^2 +
6 a^8 b^6 c^2 - 8 a^6 b^8 c^2 - a^4 b^10 c^2 + a^2 b^12 c^2 +
b^14 c^2 + 5 a^12 c^4 + 4 a^10 b^2 c^4 - 24 a^8 b^4 c^4 +
12 a^6 b^6 c^4 + 12 a^4 b^8 c^4 - 3 a^2 b^10 c^4 - 6 b^12 c^4 -
10 a^10 c^6 + 6 a^8 b^2 c^6 + 12 a^6 b^4 c^6 - 24 a^4 b^6 c^6 +
2 a^2 b^8 c^6 + 15 b^10 c^6 + 10 a^8 c^8 - 8 a^6 b^2 c^8 +
12 a^4 b^4 c^8 + 2 a^2 b^6 c^8 - 20 b^8 c^8 - 5 a^6 c^10 -
a^4 b^2 c^10 - 3 a^2 b^4 c^10 + 15 b^6 c^10 + a^4 c^12 +
a^2 b^2 c^12 - 6 b^4 c^12 + b^2 c^14,
a^12 b^4 - 5 a^10 b^6 + 10 a^8 b^8 - 10 a^6 b^10 + 5 a^4 b^12 -
a^2 b^14 + a^14 c^2 + a^12 b^2 c^2 - a^10 b^4 c^2 - 8 a^8 b^6 c^2 +
6 a^6 b^8 c^2 + 4 a^4 b^10 c^2 - 2 a^2 b^12 c^2 - b^14 c^2 -
6 a^12 c^4 - 3 a^10 b^2 c^4 + 12 a^8 b^4 c^4 + 12 a^6 b^6 c^4 -
24 a^4 b^8 c^4 + 4 a^2 b^10 c^4 + 5 b^12 c^4 + 15 a^10 c^6 +
2 a^8 b^2 c^6 - 24 a^6 b^4 c^6 + 12 a^4 b^6 c^6 + 6 a^2 b^8 c^6 -
10 b^10 c^6 - 20 a^8 c^8 + 2 a^6 b^2 c^8 + 12 a^4 b^4 c^8 -
8 a^2 b^6 c^8 + 10 b^8 c^8 + 15 a^6 c^10 - 3 a^4 b^2 c^10 -
a^2 b^4 c^10 - 5 b^6 c^10 - 6 a^4 c^12 + a^2 b^2 c^12 + b^4 c^12 +
a^2 c^14,
a^14 b^2 - 6 a^12 b^4 + 15 a^10 b^6 - 20 a^8 b^8 + 15 a^6 b^10 -
6 a^4 b^12 + a^2 b^14 + a^12 b^2 c^2 - 3 a^10 b^4 c^2 +
2 a^8 b^6 c^2 + 2 a^6 b^8 c^2 - 3 a^4 b^10 c^2 + a^2 b^12 c^2 +
a^12 c^4 - a^10 b^2 c^4 + 12 a^8 b^4 c^4 - 24 a^6 b^6 c^4 +
12 a^4 b^8 c^4 - a^2 b^10 c^4 + b^12 c^4 - 5 a^10 c^6 -
8 a^8 b^2 c^6 + 12 a^6 b^4 c^6 + 12 a^4 b^6 c^6 - 8 a^2 b^8 c^6 -
5 b^10 c^6 + 10 a^8 c^8 + 6 a^6 b^2 c^8 - 24 a^4 b^4 c^8 +
6 a^2 b^6 c^8 + 10 b^8 c^8 - 10 a^6 c^10 + 4 a^4 b^2 c^10 +
4 a^2 b^4 c^10 - 10 b^6 c^10 + 5 a^4 c^12 - 2 a^2 b^2 c^12 +
5 b^4 c^12 - a^2 c^14 - b^2 c^14}


OI:


{a^2 (a^10 b^6 - 3 a^8 b^8 + 3 a^6 b^10 - a^4 b^12 - a^8 b^6 c^2 -
a^6 b^8 c^2 + 2 a^4 b^10 c^2 + 4 a^6 b^6 c^4 - 5 a^4 b^8 c^4 +
a^10 c^6 - a^8 b^2 c^6 + 4 a^6 b^4 c^6 + a^2 b^8 c^6 - b^10 c^6 -
3 a^8 c^8 - a^6 b^2 c^8 - 5 a^4 b^4 c^8 + a^2 b^6 c^8 +
2 b^8 c^8 + 3 a^6 c^10 + 2 a^4 b^2 c^10 - b^6 c^10 - a^4 c^12),
b^2 (-a^12 b^4 + 3 a^10 b^6 - 3 a^8 b^8 + a^6 b^10 + 2 a^10 b^4 c^2 -
a^8 b^6 c^2 - a^6 b^8 c^2 - 5 a^8 b^4 c^4 + 4 a^6 b^6 c^4 -
a^10 c^6 + a^8 b^2 c^6 + 4 a^4 b^6 c^6 - a^2 b^8 c^6 + b^10 c^6 +
2 a^8 c^8 + a^6 b^2 c^8 - 5 a^4 b^4 c^8 - a^2 b^6 c^8 -
3 b^8 c^8 - a^6 c^10 + 2 a^2 b^4 c^10 + 3 b^6 c^10 - b^4 c^12),
c^2 (-a^10 b^6 + 2 a^8 b^8 - a^6 b^10 + a^8 b^6 c^2 + a^6 b^8 c^2 -
a^12 c^4 + 2 a^10 b^2 c^4 - 5 a^8 b^4 c^4 - 5 a^4 b^8 c^4 +
2 a^2 b^10 c^4 - b^12 c^4 + 3 a^10 c^6 - a^8 b^2 c^6 +
4 a^6 b^4 c^6 + 4 a^4 b^6 c^6 - a^2 b^8 c^6 + 3 b^10 c^6 -
3 a^8 c^8 - a^6 b^2 c^8 - a^2 b^6 c^8 - 3 b^8 c^8 + a^6 c^10 +
b^6 c^10)}

OK:

{a^2 (a^10 b^6 - 3 a^8 b^8 + 3 a^6 b^10 - a^4 b^12 - a^8 b^6 c^2 -
a^6 b^8 c^2 + 2 a^4 b^10 c^2 + 4 a^6 b^6 c^4 - 5 a^4 b^8 c^4 +
a^10 c^6 - a^8 b^2 c^6 + 4 a^6 b^4 c^6 + a^2 b^8 c^6 - b^10 c^6 -
3 a^8 c^8 - a^6 b^2 c^8 - 5 a^4 b^4 c^8 + a^2 b^6 c^8 +
2 b^8 c^8 + 3 a^6 c^10 + 2 a^4 b^2 c^10 - b^6 c^10 - a^4 c^12),
b^2 (-a^12 b^4 + 3 a^10 b^6 - 3 a^8 b^8 + a^6 b^10 + 2 a^10 b^4 c^2 -
a^8 b^6 c^2 - a^6 b^8 c^2 - 5 a^8 b^4 c^4 + 4 a^6 b^6 c^4 -
a^10 c^6 + a^8 b^2 c^6 + 4 a^4 b^6 c^6 - a^2 b^8 c^6 + b^10 c^6 +
2 a^8 c^8 + a^6 b^2 c^8 - 5 a^4 b^4 c^8 - a^2 b^6 c^8 -
3 b^8 c^8 - a^6 c^10 + 2 a^2 b^4 c^10 + 3 b^6 c^10 - b^4 c^12),
c^2 (-a^10 b^6 + 2 a^8 b^8 - a^6 b^10 + a^8 b^6 c^2 + a^6 b^8 c^2 -
a^12 c^4 + 2 a^10 b^2 c^4 - 5 a^8 b^4 c^4 - 5 a^4 b^8 c^4 +
2 a^2 b^10 c^4 - b^12 c^4 + 3 a^10 c^6 - a^8 b^2 c^6 +
4 a^6 b^4 c^6 + 4 a^4 b^6 c^6 - a^2 b^8 c^6 + 3 b^10 c^6 -
3 a^8 c^8 - a^6 b^2 c^8 - a^2 b^6 c^8 - 3 b^8 c^8 + a^6 c^10 +
b^6 c^10)}

The general point of intersection, for a line L = ux+vy+wz=0 is:

{-a^6 b^2 u^4 + a^4 b^4 u^4 - a^6 c^2 u^4 - 2 a^4 b^2 c^2 u^4 +
a^4 c^4 u^4 + 4 a^6 b^2 u^3 v - 4 a^4 b^4 u^3 v +
4 a^4 b^2 c^2 u^3 v - 6 a^6 b^2 u^2 v^2 + 6 a^4 b^4 u^2 v^2 +
6 a^4 b^2 c^2 u^2 v^2 + 4 a^6 b^2 u v^3 - 4 a^4 b^4 u v^3 -
8 a^4 b^2 c^2 u v^3 - 4 a^2 b^4 c^2 u v^3 + 4 a^2 b^2 c^4 u v^3 -
a^6 b^2 v^4 + a^4 b^4 v^4 + 3 a^4 b^2 c^2 v^4 + a^2 b^4 c^2 v^4 +
b^6 c^2 v^4 - 3 a^2 b^2 c^4 v^4 - 2 b^4 c^4 v^4 + b^2 c^6 v^4 +
4 a^6 c^2 u^3 w + 4 a^4 b^2 c^2 u^3 w - 4 a^4 c^4 u^3 w -
24 a^4 b^2 c^2 u^2 v w + 12 a^4 b^2 c^2 u v^2 w +
12 a^2 b^4 c^2 u v^2 w - 12 a^2 b^2 c^4 u v^2 w -
4 a^4 b^2 c^2 v^3 w - 4 b^6 c^2 v^3 w + 8 a^2 b^2 c^4 v^3 w +
8 b^4 c^4 v^3 w - 4 b^2 c^6 v^3 w - 6 a^6 c^2 u^2 w^2 +
6 a^4 b^2 c^2 u^2 w^2 + 6 a^4 c^4 u^2 w^2 +
12 a^4 b^2 c^2 u v w^2 - 12 a^2 b^4 c^2 u v w^2 +
12 a^2 b^2 c^4 u v w^2 - 6 a^2 b^4 c^2 v^2 w^2 +
6 b^6 c^2 v^2 w^2 - 6 a^2 b^2 c^4 v^2 w^2 - 12 b^4 c^4 v^2 w^2 +
6 b^2 c^6 v^2 w^2 + 4 a^6 c^2 u w^3 - 8 a^4 b^2 c^2 u w^3 +
4 a^2 b^4 c^2 u w^3 - 4 a^4 c^4 u w^3 - 4 a^2 b^2 c^4 u w^3 -
4 a^4 b^2 c^2 v w^3 + 8 a^2 b^4 c^2 v w^3 - 4 b^6 c^2 v w^3 +
8 b^4 c^4 v w^3 - 4 b^2 c^6 v w^3 - a^6 c^2 w^4 +
3 a^4 b^2 c^2 w^4 - 3 a^2 b^4 c^2 w^4 + b^6 c^2 w^4 + a^4 c^4 w^4 +
a^2 b^2 c^4 w^4 - 2 b^4 c^4 w^4 + b^2 c^6 w^4,
a^4 b^4 u^4 - a^2 b^6 u^4 + a^6 c^2 u^4 + a^4 b^2 c^2 u^4 +
3 a^2 b^4 c^2 u^4 - 2 a^4 c^4 u^4 - 3 a^2 b^2 c^4 u^4 +
a^2 c^6 u^4 - 4 a^4 b^4 u^3 v + 4 a^2 b^6 u^3 v -
4 a^4 b^2 c^2 u^3 v - 8 a^2 b^4 c^2 u^3 v + 4 a^2 b^2 c^4 u^3 v +
6 a^4 b^4 u^2 v^2 - 6 a^2 b^6 u^2 v^2 + 6 a^2 b^4 c^2 u^2 v^2 -
4 a^4 b^4 u v^3 + 4 a^2 b^6 u v^3 + 4 a^2 b^4 c^2 u v^3 +
a^4 b^4 v^4 - a^2 b^6 v^4 - 2 a^2 b^4 c^2 v^4 - b^6 c^2 v^4 +
b^4 c^4 v^4 - 4 a^6 c^2 u^3 w - 4 a^2 b^4 c^2 u^3 w +
8 a^4 c^4 u^3 w + 8 a^2 b^2 c^4 u^3 w - 4 a^2 c^6 u^3 w +
12 a^4 b^2 c^2 u^2 v w + 12 a^2 b^4 c^2 u^2 v w -
12 a^2 b^2 c^4 u^2 v w - 24 a^2 b^4 c^2 u v^2 w +
4 a^2 b^4 c^2 v^3 w + 4 b^6 c^2 v^3 w - 4 b^4 c^4 v^3 w +
6 a^6 c^2 u^2 w^2 - 6 a^4 b^2 c^2 u^2 w^2 - 12 a^4 c^4 u^2 w^2 -
6 a^2 b^2 c^4 u^2 w^2 + 6 a^2 c^6 u^2 w^2 -
12 a^4 b^2 c^2 u v w^2 + 12 a^2 b^4 c^2 u v w^2 +
12 a^2 b^2 c^4 u v w^2 + 6 a^2 b^4 c^2 v^2 w^2 -
6 b^6 c^2 v^2 w^2 + 6 b^4 c^4 v^2 w^2 - 4 a^6 c^2 u w^3 +
8 a^4 b^2 c^2 u w^3 - 4 a^2 b^4 c^2 u w^3 + 8 a^4 c^4 u w^3 -
4 a^2 c^6 u w^3 + 4 a^4 b^2 c^2 v w^3 - 8 a^2 b^4 c^2 v w^3 +
4 b^6 c^2 v w^3 - 4 a^2 b^2 c^4 v w^3 - 4 b^4 c^4 v w^3 +
a^6 c^2 w^4 - 3 a^4 b^2 c^2 w^4 + 3 a^2 b^4 c^2 w^4 - b^6 c^2 w^4 -
2 a^4 c^4 w^4 + a^2 b^2 c^4 w^4 + b^4 c^4 w^4 + a^2 c^6 w^4,
a^6 b^2 u^4 - 2 a^4 b^4 u^4 + a^2 b^6 u^4 + a^4 b^2 c^2 u^4 -
3 a^2 b^4 c^2 u^4 + a^4 c^4 u^4 + 3 a^2 b^2 c^4 u^4 - a^2 c^6 u^4 -
4 a^6 b^2 u^3 v + 8 a^4 b^4 u^3 v - 4 a^2 b^6 u^3 v +
8 a^2 b^4 c^2 u^3 v - 4 a^2 b^2 c^4 u^3 v + 6 a^6 b^2 u^2 v^2 -
12 a^4 b^4 u^2 v^2 + 6 a^2 b^6 u^2 v^2 - 6 a^4 b^2 c^2 u^2 v^2 -
6 a^2 b^4 c^2 u^2 v^2 - 4 a^6 b^2 u v^3 + 8 a^4 b^4 u v^3 -
4 a^2 b^6 u v^3 + 8 a^4 b^2 c^2 u v^3 - 4 a^2 b^2 c^4 u v^3 +
a^6 b^2 v^4 - 2 a^4 b^4 v^4 + a^2 b^6 v^4 - 3 a^4 b^2 c^2 v^4 +
a^2 b^4 c^2 v^4 + 3 a^2 b^2 c^4 v^4 + b^4 c^4 v^4 - b^2 c^6 v^4 -
4 a^4 b^2 c^2 u^3 w + 4 a^2 b^4 c^2 u^3 w - 4 a^4 c^4 u^3 w -
8 a^2 b^2 c^4 u^3 w + 4 a^2 c^6 u^3 w + 12 a^4 b^2 c^2 u^2 v w -
12 a^2 b^4 c^2 u^2 v w + 12 a^2 b^2 c^4 u^2 v w -
12 a^4 b^2 c^2 u v^2 w + 12 a^2 b^4 c^2 u v^2 w +
12 a^2 b^2 c^4 u v^2 w + 4 a^4 b^2 c^2 v^3 w -
4 a^2 b^4 c^2 v^3 w - 8 a^2 b^2 c^4 v^3 w - 4 b^4 c^4 v^3 w +
4 b^2 c^6 v^3 w + 6 a^4 c^4 u^2 w^2 + 6 a^2 b^2 c^4 u^2 w^2 -
6 a^2 c^6 u^2 w^2 - 24 a^2 b^2 c^4 u v w^2 +
6 a^2 b^2 c^4 v^2 w^2 + 6 b^4 c^4 v^2 w^2 - 6 b^2 c^6 v^2 w^2 -
4 a^4 c^4 u w^3 + 4 a^2 b^2 c^4 u w^3 + 4 a^2 c^6 u w^3 +
4 a^2 b^2 c^4 v w^3 - 4 b^4 c^4 v w^3 + 4 b^2 c^6 v w^3 +
a^4 c^4 w^4 - 2 a^2 b^2 c^4 w^4 + b^4 c^4 w^4 - a^2 c^6 w^4 -
b^2 c^6 w^4}

Francisco Javier García Capitán
12 December 2011

NINE POINT CIRCLE

Let ABC be a triangle, A'B'C' the orthic triangle and P a point.

Let A*,B*,C* be the orthogonal projections of A,B,C on the line OP, resp.
Let L1,L2,L3 be the reflections of A'A*,B'B*,C'C* in the altitudes AA',BB',CC', resp. and M1,M2,M3 the parallels through A,B,C, to L1,L2,L3, resp.

The lines M1,M2,M3 concur at a point Q on the Nine Point Circle of ABC (Q is the center of the rectangular circumhyperbola which is the isogonal conjugate of the line OP)

APH, 9 December 2011

LOCUS

Generalization of Hyacinthos Message 10485

Let ABC be a triangle Q1, Q2 two fixed points and P a variable point. Let L1,L2,L3 be the parallels through P to AQ2, BQ2, CQ2, respectively.

Ab := L2 /\ (Parallel to BQ1 through A)
Ac := L3 /\ (Parallel to CQ1 through A)

Similarly:

Bc := L3 /\ (Parallel to CQ1 through B)
Ba := L1 /\ (Parallel to AQ1 through B)

Ca := L1 /\ (Parallel to AQ1 through C)
Cb := L2 /\ (Parallel to BQ1 through C)

Which is the locus of P such that the Euler Lines of AAbAc, BBcBa, CCaCb are concurrent?

APH, 9 December 2011