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such that:
for all 
be an acute-angled scalene triangle with circumcenter 
. Denote by 
, 
, and 
the midpoints of sides 
, 
, and 
, respectively. Let 
be the circle passing through 
and tangent to 
at . The circle intersects and 
at points 
and 
, respectively (where and are distinct from ). Let 
be the midpoint of segment 
, and let 
be the intersection of lines and 
. Prove that 
and that triangle 
is isosceles.
chessboard has a real number written in it in such a way that each number is equal to the geometric mean of all the numbers a knight's move away from it.
be a parallelogram but not a rhombus. Let 
be the foot of the perpendicular from 
to 
. The line through which is perpendicular to 
intersects 
at 
and 
at 
. Show that 
if, and only if, is a rectangle.
and 
Prove that
