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similar triangles with integer sides, and 2 equal sides

Source: Netherlands - Dutch NMO 1968 p3

February 1, 2023
geometrysimilar triangles

Problem Statement

△ABC∼△A′B′C′\vartriangle ABC \sim \vartriangle A'B'C'. △ABC\vartriangle ABC has sides a,b,ca,b,c and △A′B′C′\vartriangle A'B'C' has sides a′,b′,c′a',b',c'. Two sides of △ABC\vartriangle ABC are equal to sides of △A′B′C′\vartriangle A'B'C'. Furthermore, a<a′a < a', a<b<ca < b < c, a=8a = 8. Prove that there is exactly one pair of such triangles with all sides integers.
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