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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)
be a trapezoid with 
.
and 
be similar isosceles triangles with 
and 
erected externally to 
is perpendicular to 
and 
is perpendicular to 
.
Y by Adventure10, Mango247
Let 
be an equilateral triangle and let 
be a point in its interior. Let the lines 
, 
, 
meet the sides 
, 
, 
at the points 
, 
, 
, respectively. Prove that
.
be an equilateral triangle and let 
be a point in its interior. Let the lines 
, 
, 
meet the sides 
, 
, 
at the points 
, 
, 
, respectively. Prove that
.
etc so multiply together
Now, from the Law of Cosines, 
From the trivial inequality, 
Similarly, we find that 
Multiplying these inequalities together, we obtain 
as desired.
But 
but this is evident as the angles sum to 
.
