Y by buratinogigle
Let
be an acute-angled, non-isosceles triangle with circumcenter
and incenter
, such that
![\[
\prod_{\text{cyc}} \left( \frac{1}{a+b-c} + \frac{1}{a+c-b} - \frac{2}{b+c-a} \right) \neq 0,
\]](//latex.artofproblemsolving.com/7/8/1/78125759bf5c73149e4a56538e9b0c44309704a7.png)
where
,
, and
.
Let
,
, and
be the excenters opposite to vertices
,
, and
, respectively, and let
,
, and
be the Lemoine points of triangles
,
, and
, respectively.
Prove that the circles
,
, and
all pass through a common point
. Moreover, the isogonal conjugate of
with respect to
lies on the line
.



![\[
\prod_{\text{cyc}} \left( \frac{1}{a+b-c} + \frac{1}{a+c-b} - \frac{2}{b+c-a} \right) \neq 0,
\]](http://latex.artofproblemsolving.com/7/8/1/78125759bf5c73149e4a56538e9b0c44309704a7.png)
where



Let












Prove that the circles







This post has been edited 1 time. Last edited by Ktoan07, Apr 18, 2025, 10:26 AM