AoPS Academy
, inscribed in the circle 
, satisfies 
and 
, and 
is the intersection point of the diagonals 
and 
. The circle with center 
and radius 
intersects in two points 
and 
. Prove that the line 
is tangent to the circles with diameters 
and 
.
satisfying the equation![\[
(f(x+y))^2= f(x^2) + f(2xf(y) + y^2), \quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/2/5/e/25eed911f303ed7cc4e03765a2f943ad4641c451.png)
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point 
. Let 
be a point on the arc (not containing ) such that 
and 
. Let the lines 
and 
intersect at point 
. Let 
be the reflection of with respect to the line . The lines 
and 
intersect at point 
, and intersects the circumcircle of again at point 
. lies on the circumcircle of 
.
be non-negative real numbers such that 
Prove that
with center 
and 
, are given in the plane. We also have 
and 
. If is the midpoint of 
and the midpoint of 
, prove that the triangles 
and 
are similar.
is a 30-60-90 triangle with 
. Extend 
to 
such that is the midpoint of . Extend 
to such that 
is the midpoint of 
. Then 
are clearly 30-60-90 triangles, so by spiral symmetry, 
. But since 
, we have 
and 
. Thus 
and 
, so is a 30-60-90 triangle. Similarly for 
, and both are similar, so we are done.
