AoPS Academy
be a trapezoid with 
parallel to 
. Let 
and 
be similar isosceles triangles with 
and 
erected externally to . Let 
be the point such that 
is perpendicular to 
and 
is perpendicular to 
. Prove 
that 
and 
. Prove that from the lengths , 
and 
, you can make a right triangle.
. From the given conditions 
. Choose the point 
on the bisector of angle 
such that it lies in the halfplane determined by not containing 
and 
. Then it is pretty well-known that the triangles 
and 
are equilateral. Thus
and 
. Hence the triangle 
has sidelengths equal to 
and is right-angled.
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,![\[\frac{ac}{1-b}+\frac{bd}{1-c}+\frac{ce}{1-d}+\frac{da}{1-e}+\frac{eb}{1-a}\leq \frac{1}{4}\]](http://latex.artofproblemsolving.com/c/b/3/cb30518369e2a5f69e21233936e9a0175302645f.png)
![\[\frac{ab}{1-c}+\frac{bc}{1-d}+\frac{cd}{1-e}+\frac{de}{1-a}+\frac{ea}{1-b}\leq \frac{8}{27}\]](http://latex.artofproblemsolving.com/8/a/8/8a886552138b9e6f65a7f85266af67bf26bdf5c2.png)
