HYACINTHOS 6835

Dear Clark,

There are two easy ways of obtaining polynomial centers on the incircle.

(1) If P is a polynomial center of center on the incircle, and Q is an
arbitrary polynomial center, then the second intersection of the line PQ and
the incircle is a polynomial center.

(2) Every point on the Steiner in-ellipse is the barycentric square of an
infinite point. Every point on the incircle is the barycentric product of the
Gergonne point and a point on the Steiner in-ellipse. It follows that if
(u:v:w) is an infinite point (with polynomial coordinates), then
((s-b)(s-c)u^2 : (s-c)(s-a)v^2 : (s-a)(s-b)w^2)
is a point (polynomial center) on the incircle.

The Feuerbach point, for example, arises from the infinite point
((b-c)(s-a):(c-a)(s-b):(a-b)(s-c)).

From the ``simplest'' infinite point (b-c:c-a:a-b), we obtain
the polynomial center

((s-b)(s-c)(b-c)^2: (s-c)(s-a)(c-a)^2 : (s-a)(s-b)(a-b)^2).

I have compiled a list of some 30 such points with coordinates involving
polynomials of degrees not more than 3. When I go to office on Monday, I shall
be able to forward the list to you.

Best regards
Sincerely
Paul Yiu