# Inradius of Heron triangles

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Find all Heron triangles with inradius 1, then all heron triangles with inradius 3/2.

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This solution explains how to answer a question regarding Heron triangle.

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There are infinitely many Heronion triangles, and any one of them can be scaled by an appropriate factor to produce a new Heronian triangle with inradius 1 or 3/2, as desired.

Suppose (a, b, c) is a Heronion triangle with area A and semiperimeter s.

Then A, a, b, c, and s are rational with A^2 = s(s-a)(s-b)(s-c).

Claim: Scaling all side lengths by f = s/A results in a Heronion triangle with inradius 1.

Proof: Clearly all side lengths, and the semiperimeter, are rational after being scaled by the rational number f. The area of the scaled triangle is also rational, since ...

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