AoPS Academy
be a cyclic quadrilateral. Let 
be the midpoint of the arc 
of its circumcircle which does not contain 
or 
. Let the lines 
and 
meet at 
and the lines 
and 
meet at 
. Prove that the lines 
and are parallel.
of the triangle 
is extended beyond to so that 
. The side 
is extended beyond 
to 
so that 
. Prove that, if 
, then the triangle is right-angled.
be circumscribed about circle 
, and let 
be the orthocenter of 
. The circle touches line at . The tangent to the circle 
at meets at 
. Let 
be the midpoint of 
, and let the line 
meet again at 
. The tangent to parallel to meets the line 
at 
. Prove that 
is tangent to .
of corresponding altitude. That will give us triangle 
and our points lie one on each side of this triangle. Now it took me some time but eventually I calculated that these points has to split those sides such that 
in order for triangle 
to have required area. Now it wasn't obvious to me why triangles 
have the same area, I just haven't done those problems in a while I guess so I calculated that shortly with using coordinates.
and six non-intersecting segments from the following 
divide the triangle 
