2021 WSMO Speed Round Problems/Problem 7
Problem
Consider triangle with side lengths
and incircle
. A second circle
is drawn which is tangent to
and externally tangent to
. The radius of
can be expressed as
, where
and
is not divisible by the square of any prime. Find
.
Solution
Note that the length of the -angle bisector is
Now, let
be the incenter of triangle
This means that
Now, from Heron's formula, we find that the area of triangle
is
Since the area of a triangle is the product of the semi-perimeter and the inradius, we find that the length of the inradius of triangle
is
Now, let the radius of
be
From similar triangles, we find that
~pinkpig