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is not a perfect square, then there exists an odd prime 
s.t. is a quadratic non-residue mod .
be the orthocenter of the triangle 
. Let 
and 
be the midpoints of the sides 
and 
, respectively. Assume that lies inside the quadrilateral 
and that the circumcircles of triangles 
and 
are tangent to each other. The line through parallel to 
intersects the circumcircles of the triangles and in the points 
and 
, respectively. Let 
be the intersection point of 
and 
and let 
be the incenter of triangle 
. Prove that 
.
that satisfies 
for all real 
triangle with 
. We take , point on AC, AB respectively such that 
, 
. 
, 
lines intersect at point . If 
, find 
