Mock AIME 6 2006-2007 Problems/Problem 9
Problem
is a triangle with integer side lengths. Extend beyond to point such that . Similarly, extend beyond to point such that and beyond to point such that . If triangles , , and all have the same area, what is the minimum possible area of triangle ?
Solution
Let , , and , be the lengths of sides , and respectively.
Let , , and , be the heights of from sides , and respectively.
Since the areas of triangles , , and are equal, then,
Therefore,
and
Since the area of is half any base times it's height, then:
Therefore,
and
Since , , and , are integers, and is a prime number, then the minimum integer value that can have in order for and to also be integer is
Therefore , , and
minimum possible area of triangle using Heron's formula is is:
, where
~Tomas Diaz. orders@tomasdiaz.com
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.