be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter 
and orthocenter 
.
is constant.
be tangent to the side 
.
on 
and 
on 
and 
,
lies on the circumcircle of triangle 
is a function for which the expression 
for all 
is always a perfect square. Prove that 
for all 
.
and 
,
.
Y by
Given an acute-angled non-isosceles triangle 
, point 
is the center of its circumscribed circle. The extension of altitude 
of triangle intersects it's circumscribed circle at point 
. On the sides 
and 
the points 
and 
are marked, respectively, such that 
and 
. The circumscribed circle of the triangle 
intersects the segment at the point 
. Prove that 
.
, point 
is the center of its circumscribed circle. The extension of altitude 
of triangle intersects it's circumscribed circle at point 
. On the sides 
and 
the points 
and 
are marked, respectively, such that 
and 
. The circumscribed circle of the triangle 
intersects the segment at the point 
. Prove that 
.
