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    How to Find 5 Triangles with Integer Sides

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    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.

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    Solution Preview

    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 = ...

    Solution Summary

    The expert determines how to find 5 triangles with integer sides.