Canarina canariensis

Re: RADICAL CENTERS - NPC - OI LINE

Posted By: amontes1949

Wed Aug 21, 2013 4:44 am

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and A'B'C' the cevian triangle of I.

Denote:

(Nab), (Nac) = the NPCs of AIB', AIC', resp.

(Nbc), (Nba) = the NPCs of BIC', BIA', resp.

(Nca), (Ncb) = the NPCs of CIA', CIB', resp.

R = the radical center of (Nbc), (Nca), (Nab)

S = the radical center of (Nba), (Ncb), (Nac)

*** (Trilinear ccordinates)

R = ((a + b - c) (a - b + c) (a^3 b - a b^3 + 2 a^3 c - 2 a b^2 c - b^3 c + a^2 c^2 - 3 a b c^2 - 3 b^2 c^2 - 2 a c^3 - 3 b c^3 - c^4) : (a - b - c) (a + b - c) (a^4 + 2 a^3 b - a^2 b^2 - 2 a b^3 + 3 a^3 c + 3 a^2 b c - b^3 c + 3 a^2 c^2 + 2 a b c^2 + a c^3 + b c^3) : (a - b - c) (a - b + c) (a^3 b + 3 a^2 b^2 + 3 a b^3 + b^4 + a^3 c + 2 a^2 b c + 3 a b^2 c + 2 b^3 c - b^2 c^2 - a c^3 - 2 b c^3)) T = ((a + b - c) (a - b + c) (2 a^3 b + a^2 b^2 - 2 a b^3 - b^4 + a^3 c - 3 a b^2 c - 3 b^3 c - 2 a b c^2 - 3 b^2 c^2 - a c^3 - b c^3): (a - b - c) (a + b - c) (a^3 b - a b^3 + a^3 c + 2 a^2 b c - 2 b^3 c + 3 a^2 c^2 + 3 a b c^2 - b^2 c^2 + 3 a c^3 + 2 b c^3 + c^4) : (a - b - c) (a - b + c) (a^4 + 3 a^3 b + 3 a^2 b^2 + a b^3 + 2 a^3 c + 3 a^2 b c + 2 a b^2 c + b^3 c - a^2 c^2 - 2 a c^3 - b c^3) )

R and S is a bicentric pair, then the midpoint of RS is a triangle center of trilinear ccordinates:

M = ( (a + b - c) (a - b + c) (3 a^3 b + a^2 b^2 - 3 a b^3 - b^4 + 3 a^3 c - 5 a b^2 c - 4 b^3 c + a^2 c^2 - 5 a b c^2 - 6 b^2 c^2 - 3 a c^3 - 4 b c^3 - c^4) : (a - b - c) (a + b - c) (a^4 + 3 a^3 b - a^2 b^2 - 3 a b^3 + 4 a^3 c + 5 a^2 b c - 3 b^3 c + 6 a^2 c^2 + 5 a b c^2 - b^2 c^2 + 4 a c^3 + 3 b c^3 + c^4) : (a - b - c) (a - b + c) (a^4 + 4 a^3 b + 6 a^2 b^2 + 4 a b^3 + b^4 + 3 a^3 c + 5 a^2 b c + 5 a b^2 c + 3 b^3 c - a^2 c^2 - b^2 c^2 - 3 a c^3 - 3 b c^3) )

M lie on the central line X(1)X(3).

Ideal point of line RT is X(513).

This bicentric pair does not appear in the current edition of "BICENTRIC PAIRS OF POINTS" ( Clark Kimberling)

I suggest as a flower name for this bicentric pair: Canarina

Angel Montesdeoca

Anopolis #860