HYACINTHOS 8070

Dear Hyacinthians,

in August 2003 a cluster of new points were added to the ETC thanks
to Peter Moses.
Some of these points were related to the Apollonius-circle
(see Forum Geometricorum 2002 p. 175-182 by Darij Grinberg and Paul
Yiu)

I call the triangle formed by the points F'a, F'b and F'c where the
Apollonius-circle is tangent to the excircles the Apollonius-
triangle.

I have proved, by barycentic computations, that the Apollonius
triangle and the cevian triangle of the Lemoine point are
perspective.

The barycentric coordinates of the perspector are

( aa(b+c)(a(b+c)+bb+cc) :
bb(c+a)(b(c+a)+cc+aa) :
cc(a+b)(c(a+b)+aa+bb) )

It's search-value should be 1.427 222 829 750
I couldn't find it in the ETC

Greetings from Bruges

Eric Danneels