--- In

Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@...> wrote:

>

> For which P's the NPC of PBC is tangent to the circumcircle of ABC?

>

> APH

>

Dear Antreas

It is a quartic (cA) through B and C (cusps).

Equation of (cA) (García Capitán,

http://tech.groups.yahoo.com/group/Hyacinthos/message/21696):

a^8 x^4 - 2 a^4 b^4 x^4 + b^8 x^4 - 2 a^4 c^4 x^4 - 2 b^4 c^4 x^4 +

c^8 x^4 + 4 a^8 x^3 y - 4 a^6 b^2 x^3 y - 4 a^4 b^4 x^3 y +

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

4 a^8 x^2 y^2 - 8 a^6 b^2 x^2 y^2 + 4 a^4 b^4 x^2 y^2 +

4 a^8 x^3 z - 4 a^4 b^4 x^3 z - 4 a^6 c^2 x^3 z -

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

12 a^8 x^2 y z - 8 a^6 b^2 x^2 y z - 4 a^4 b^4 x^2 y z -

8 a^6 c^2 x^2 y z - 4 a^4 c^4 x^2 y z + 8 a^8 x y^2 z -

8 a^6 b^2 x y^2 z + 4 a^8 x^2 z^2 - 8 a^6 c^2 x^2 z^2 +

4 a^4 c^4 x^2 z^2 + 8 a^8 x y z^2 - 8 a^6 c^2 x y z^2 + 4 a^8 y^2 z^2 =0.

The tangent ((a^2 - b^2)x + a^2z=0) in the cusp B meet again (cA) in Ab, and the tangent ((a^2 - c^2)x + a^2y=0) in the cusp C meet again (cA) in Ac.

Define Ba, Bc, Ca, and Cb simillary, then the triangle bounded by (AbAc, BcBa, CaCb) is perspective with ABC; the perspector is:

(SA/((a^4 - 2a^2b^2 + b^4 + 2a^2b*c - 2b^3c -

2a^2c^2 + 4b^2c^2 - 2b*c^3 + c^4)*

(a^4 - 2a^2b^2 + b^4 - 2a^2b*c + 2b^3*c -

2a^2c^2 + 4b^2c^2 + 2b*c^3 + c^4)): ... : ....).

Or:

( SA/( 4a^6(b^2+c^2) - 6a^4(b^2+c^2)^2+ 4a^2(b^2+c^2)^3

-(a^8 + b^8+ c^8 +2b^2c^2(2b^2+c^2)(b^2+2c^2)) ): ... : ...).

With (6-9-13)-search number -0.688951377609844758609146059...

Best regards

Angel Montesdeoca