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 FaThEr-SqUiRrEl
In an acute triangle 
, points 
and 
are respectively the midpoints of the sides 
and 
. The altitude of triangle from vertex 
intersects the segment 
at point 
. The perpendicular bisector of 
intersects the segment at point 
. Show, that 
.
, points 
and 
are respectively the midpoints of the sides 
and 
. The altitude of triangle from vertex 
intersects the segment 
at point 
. The perpendicular bisector of 
intersects the segment at point 
. Show, that 
.
be the midpoint of 
.
is the midpoint of 
,
congruent to 
,
,
