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such that for any 
:
such that 
for all 
be a triangle with circumcircle 
and let 
be the 
-excircle. Let 
and 
be the intersection points of and . Let 
and 
be the projections of onto the tangent lines to at and respectively. The tangent line at to the circumcircle of the triangle 
intersects the tangent line at to the circumcircle of the triangle 
at a point 
. Prove that 
.
and 
be isosceles triangle with the base 
. We know that 
. Let 
be the midpoint of . The points 
are chosen such that 
, 
, 
, and 
. Prove that 
is the midpoint of 
and 
.
the point of intersection of the lines ED, BC .
so the triangles BDE and CDF are congruent.
.
so the triangles EDP and PDF are congruent . So 
so
and 
, 
, 
(a.s.a.) 
Apply the Menelaus' theorem to the transversal 
for the triangle :
the triangle 
is isosceles in the vertex , 
is the bisector of the angle 
![\[
\begin{cases}
x - y z = 1,\\[2pt]
y - z x = 2,\\[2pt]
z - x y = 4.
\end{cases}
\]](http://latex.artofproblemsolving.com/b/2/d/b2d94e49491a71f86109103f985fe3f238cf4d95.png)
and 
we get: 
,
and 
,
and 
we get:
.