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be a cyclic quadrilateral. The point 
is the reflection of 
the intersection of 
and 
, the point 
is the reflection of midpoint of 
. Also let 
be the reflection of 
midpoint of 
. Show that 
concyclic.
. Prove that: 
be a triangle, and let 
be a point on side . A line through intersects side 
at 
and ray 
at 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle again at point 
distinct from point . The lines 
and 
intersect again at 
and 
respectively.
such that 
for all real number 
and 
and 
) equals three times the sum of the areas of the three inner squares (
and 
).
we construct outwardly the squares 
Let 
be the areas of the squares with side length 
Prove that![\[[ABED] + [CHFB] + [ALKC]= \frac{_1}{^3}(S_1 + S_2 + S_3) \]](http://latex.artofproblemsolving.com/1/9/a/19a9c0aa6f1eeaa0a509e8a57a66a4d22f70b50e.png)
be the reflection of 
about 
Then it is easy to see that 
and 
are congruent. Thus if 
is the midpoint of 
we have 
This means that 
are twice the corresponding medians 
of 
From 
we deduce then the desired relation.
