The Short, Unhappy Life Of A Doomed Conjecture


So last month amongst the talk about the radius of a circle inscribed in a Pythagorean right triangle I mentioned that I had, briefly, floated a conjecture that might have spun off it. It didn’t, though I promised to describe the chain of thought I had while exploring it, on the grounds that the process of coming up with mathematical ideas doesn’t get described much, and certainly doesn’t get described for the sorts of fiddling little things that make up a trifle like this.

A triangle with sides a, b, and c, and an inscribed circle. From the center of the circle are lines going to the vertices of the triangle, dividing the circle into three smaller triangles, with bases of lengths, a, b, and c respectively and all with the same height, r, the radius of the inscribed circle.
A triangle (meant to be a right triangle) with an inscribed circle of radius r. The triangle is divided into three smaller triangles meeting at the center of the inscribed circle.

The point from which I started was a question about the radius of a circle inscribed in the right triangle with legs of length 5, 12, and 13. This turns out to have a radius of 2, which is interesting because it’s a whole number. It turns out to be simple to show that for a Pythagorean right triangle, that is, a right triangle whose legs are a Pythagorean triple — like (3, 4, 5), or (5, 12, 13), any where the square of the biggest number is the same as what you get adding together the squares of the two smaller numbers — the inscribed circle has a radius that’s a whole number. For example, the circle you could inscribe in a triangle of sides 3, 4, and 5 would have radius 1. The circle inscribed in a triangle of sides 8, 15, and 17 would have radius 3; so does the circle inscribed in a triangle of sides 7, 24, and 25.

Since I now knew that (and in multiple ways: HowardAt58 had his own geometric solution, and you can also do this algebraically) I started to wonder about the converse. If a Pythagorean right triangle’s inscribed circle has a whole number for a radius, can does knowing a circle has a whole number for a radius tell us anything about the triangle it’s inscribed in? This is an easy way to build new conjectures: given that “if A is true, then B must be true”, can it also be that “if B is true, then A must be true”? Only rarely will that be so — it’s neat when it is — but we might be able to patch something up, like, “if B, C, and D are all simultaneously true, then A must be true”, or perhaps, “if B is true, then at least E must be true”, where E resembles A but maybe doesn’t make such a strong claim. Thus are tiny little advances in mathematics created.

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