be a convex hexagon, such that the triangles 
are equilateral and the diagonals 
and 
are concurrent. Prove that 
or 
to the right;
squares north of the initial square after some finite number of moves if it is initially facing:
satisfying 
is a perfect square
,![\[
\frac{a}{b} + \frac{b}{c} + \frac{c}{a}.
\]](http://latex.artofproblemsolving.com/3/2/b/32b41635c159921e11245369e66185ec69b0d785.png)
Y by
Let 
be a triangle with area 
. Let 
be a point on segment 
. Let points 
and 
on 
and 
, respectively, satisfy 
and 
. Compute, with proof, the area of the quadrilateral with vertices at , , the midpoint of 
, and the midpoint of 
.
be a triangle with area 
. Let 
be a point on segment 
. Let points 
and 
on 
and 
, respectively, satisfy 
and 
. Compute, with proof, the area of the quadrilateral with vertices at , , the midpoint of 
, and the midpoint of 
.
.
,
into 
,
,
and split 
into 
,
,
.
