AoPS Academy
be a scalene triangle with orthocenter 
and circumcenter 
. Denote by 
, 
the midpoints of 
, 
. Suppose the circle 
with diameter meets the circumcircle of at 
, and meets line 
at a point 
. The tangent to at 
meets line 
at 
. Show that the circumcircles of 
and 
intersect at a point 
on 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of square numbers. Prove that: 
is the sum of square numbers
. Points 
are the centers of the circumscribed circles of triangles 
, 
, 
, , respectively. It is known that 
is a convex quadrilateral whose three sides are equal. Prove that is a rhombus.
. Points are the centers of the circumscribed circles of triangles , , , , respectively. It is known that is a convex quadrilateral whose three sides are equal. Prove that is a rhombus.
