Find five triangles with integers sides (without no commun divisors ) such that the median AD from angle A to side BC is m(a) = b + c - a where a, b , c are sides opposite to angles A, B, C respectively. The inner circle is tangent at point X to segment BD on side BC.
So find the 5 triangles of integers sides (with no commun divisor ) such that the length of median AD is m(a) = b + c - a.
For each of the five triangles, give separately the length of sides a, b, and c.
Call the side lengths a, b, and c, and the median m. Note that the sentence "The inner circle is tangent at point X to segment BD on side BC" has no bearing on the problem, since X is not mentioned again.
We will use Apollonius' Theorem for the length of a median (http://en.wikipedia.org/wiki/Apollonius%27_theorem):
c^2 + b^2 = ...
The expert determines how to find 5 triangles with integer sides.