AoPS Academy
be a triangle with orthocenter 
and let 
be the midpoint of 
Suppose that 
and 
are distinct points on the circle with diameter 
different from 
such that lies on line 
Prove that the orthocenter of 
lies on the circumcircle of 
, otherwise we'll look for nonconstant solutions.
for some 
:
, so 
is constant, contradiction. for some 
we must have 
and 
.
now, then 
:
for all , but since we found a that satisfies we can also conclude . Therefore 
for all 
which satisfies the equation (and the second condition).
.
.
Where 
