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AB is a square of an integer

Source: Mediterranean MO 2006

October 30, 2010
geometrysimilar trianglesgeometry proposed

Problem Statement

The side lengths a,b,ca,b,c of a triangle ABCABC are integers with gcd⁡(a,b,c)=1\gcd(a,b,c)=1. The bisector of angle BACBAC meets BCBC at DD.
(a) show that if triangles DBADBA and ABCABC are similar then cc is a square.
(b) If c=n2c=n^2 is a square (n≥2)(n\ge 2), find a triangle ABCABC satisfying (a).
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