Inradius of Heron triangles

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 the area will scale by f^2, a rational number. Therefore the scaled triangle (af, bf, cf), with area f^2*A and semiperimeter sf, is Heronion.

The inradius, r, of any triangle is given by r = area/semiperimeter. For the scaled triangle, we have

inradius
= area/semiperimeter
= f^2*A/(sf)
= fA/s
= (s/A)(A/s)
= 1

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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 = 3s/(2A) results in a Heronion triangle with inradius 3/2.

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 the area will scale by f^2, a rational number. Therefore the scaled triangle (af, bf, cf), with area f^2*A and semiperimeter sf, is Heronion.

The inradius, r, of any triangle is given by r = area/semiperimeter. For the scaled triangle, we have

inradius
= area/semiperimeter
= f^2*A/(sf)
= fA/s
= (3s/(2A))(A/s)
= 3/2

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Now that we've proven that every Heronion triangle can be scaled by an appropriate factor to produce a new Heronian triangle with inradius 1 or 3/2, we must define the infinite set of Heronion triangles. The derivation is a research-level question beyond the scope of this question, but you can see a parameterization of all Heronion triangles here:

http://mathworld.wolfram.com/HeronianTriangle.html

That gives you all Heronion triangles with integer side lengths, which can be scaled by any rational number to get all Heronion triangles. Any Heronion triangle scaled by the factors shown above gives a Heronion triangles with the desired inradius.

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