Find all Heron triangles with inradius 1, then all heron triangles with inradius 3/2.© BrainMass Inc. brainmass.com October 10, 2019, 5:08 am ad1c9bdddf
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 ...
This solution explains how to answer a question regarding Heron triangle.