.
.![\[\sum ab +\sum ab^2 \le 6\]](http://latex.artofproblemsolving.com/9/f/6/9f632ab1e6df5029bcd4416eb06623123ad2cd8c.png)
be a triangle with side‐lengths 
,
,
.
be the areas of triangles 
,
,
,![\[
\frac{abc}{12R}
\;\le\;
\frac{P_1^2 + P_2^2 + P_3^2}{P}
\;\le\;
\frac{3R^3}{4\sqrt[3]{abc}}.
\]](http://latex.artofproblemsolving.com/1/3/9/139a3e15f08afb78ca0b8301f24d30db8f202207.png)
the circle with diameter 
,
be the contact points of the tangents from 
to 
and 
lie on opposite sides of 
and 
and 
lie on opposite sides of 
.
be the circle with diameter 
,
,
the circle with diameter 
.
are concurrent.
Y by kiyoras_2001, fAaAtDoOoG
Triangle 
has side lengths 
, 
, and 
. Select a point 
inside , and construct the incenters of 
, 
, and 
and denote them as 
, 
, 
. What is the maximum area of the triangle 
?
has side lengths 
, 
, and 
. Select a point 
inside , and construct the incenters of 
, 
, and 
and denote them as 
, 
, 
. What is the maximum area of the triangle 
?
