HYACINTHOS 8076

Dear Eric,

[ED]: 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

I have just confirmed your result. Indeed the same is true for the cevian
triangle of the incenter,
and the perspector is

X(1682) = (a^2(b+c-a)(a(b+c)+(b^2+c^2)^2 : ... : ...)

which happens also to be the internal center of similitude of the incircle
and the Apollonius circle.

Best regards
Sincerely
Paul Yiu

>Greetings from Bruges
>
>Eric Danneels