Τετάρτη, 14 Δεκεμβρίου 2011

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

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