Mock AIME 2 2006-2007 Problems/Problem 14
Problem
In triangle ,
and
. Given that
,
and
intersect at
and are an angle bisector, median, and altitude of the triangle, respectively, compute the length of
Solution
Let .
By the Angle Bisector Theorem, .
Let . Then by the Pythagorean Theorem,
and
. Subtracting the former equation from the latter to eliminate
, we have
so
. Since
,
. We can solve these equations for
and
in terms of
to find that
and
.
Now, by Ceva's Theorem, , so
and
. Plugging in the values we previously found,
so
and
which yields finally .
Problem Source
4everwise thought of this problem after reading the first chapter of Geometry Revisited.