HYACINTHOS 21290

Dear Hyacinthists.

(Paul Yiu, Introduction to the Geometry of the Triangle, 2002; 12.4 The Soddy hyperbolas, p. 143)

Given triangle ABC, consider the hyperbola passing through A, and with foci at B and C. We shall call this the A-Soddy hyperbola of the triangle.

The equation of A-Soddy hyperbola is
(Fa): (c+a-b)(a+b-c)(y^2+z^2)-2(a^2+(b-c)^2)y z+4(b-c)c x y -4b(b-c)z x=0.

The perspector of a (Fa): Pa = ( SB SC : - b c SC : -b c SB ).

The polar of Pa with respect to (Fa) is is the line "da":
2b(b-c)cx + (a^2+(b-c)^2)cy - b(a^2+(b-c)^2)z=0.

Similarly define the lines "db" and "dc"; then the lines da, db and dc are concurrent at the triangle center X(4319).



Sincerely
Angel Montesdeoca