ISL 2002 G2
by Wolstenholme, Aug 3, 2014, 5:07 AM
Let 
be a triangle for which there exists an interior point 
such that 
. Let the lines 
and 
meet the sides 
and 
at 
and 
respectively. Prove that ![\[ AB+AC\geq4DE. \]](//latex.artofproblemsolving.com/3/a/4/3a49c5f8bd2930adca7bde2601dd045d4bf05ff6.png)
Proof:
It is clear that
is the Fermat point of 
so constructing points 
such that triangles 
and 
are equilateral and have interiors disjoint from the interior of , we have that 
.
Now it is clear that quadrilateral
is cyclic so the area of 
is maximal when is the midpoint of minor arc 
of the circumcircle of quadrilateral . In this case its area is one-third that of so 
. Similarly 
.
Since
are collinear and 
are collinear, this implies that 
so it suffices to show that 
.
But
by the Triangle Inequality so we are done.
be a triangle for which there exists an interior point 
such that 
. Let the lines 
and 
meet the sides 
and 
at 
and 
respectively. Prove that ![\[ AB+AC\geq4DE. \]](http://latex.artofproblemsolving.com/3/a/4/3a49c5f8bd2930adca7bde2601dd045d4bf05ff6.png)
Proof:
It is clear that
is the Fermat point of 
so constructing points 
such that triangles 
and 
are equilateral and have interiors disjoint from the interior of , we have that 
.Now it is clear that quadrilateral
is cyclic so the area of 
is maximal when is the midpoint of minor arc 
of the circumcircle of quadrilateral . In this case its area is one-third that of so 
. Similarly 
.Since
are collinear and 
are collinear, this implies that 
so it suffices to show that 
.But
by the Triangle Inequality so we are done.
August 2016
June 2016