AoPS Academy
be a scalene triangle with point 
and 
on the plane such that 
is an equilateral . Let 
intersect 
and 
at 
and 
respectively and 
intersect 
and 
at 
and 
respectively .
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of square numbers. Prove that: 
is the sum of square numbers
real number on the board such that each number is more than 
.
on the board and write 
.
of corresponding altitude. That will give us triangle 
and our points lie one on each side of this triangle. Now it took me some time but eventually I calculated that these points has to split those sides such that 
in order for triangle 
to have required area. Now it wasn't obvious to me why triangles 
have the same area, I just haven't done those problems in a while I guess so I calculated that shortly with using coordinates.
and six non-intersecting segments from the following 
divide the triangle 
