Dissect square into triangles of same perimeter

Question

Given the unit square, it is clearly possible to dissect it into any even number of triangles of the same perimeter (the triangles being even isometric). But I wasn't able to find such a dissection for 3 triangles (or 5...). Is it possible?

Answer 1

There is a famous theorem by Monsky saying that you cannot partition a square into an odd number of triangles of equal area. The proof is not easy, and some authors even take recurse to the axiom of choice for a proof. Anyway: To obtain a theorem about an odd number of triangles of equal perimeter should be even more difficult since square roots are entering the picture.

Answer 2

UPDATE: A rectangle 1 by 0.964394 (or $4 - 22x + 67x^2 - 112x^3 + 64x^4=0$) can be divided into seven triangles with equal perimeter.

I did some looking for 5, with various base triangles, and then ellipses on the unpaired edges to assure equal perimeters. Nothing obvious popped out. Here's the closest I got, with seven equal perimeter triangles almost completing the square. Point H has an $x$ value of $(10 - 3 \sqrt5)/10$.

In Wheel Graphs with Integer Edges, I give examples where integer-sided triangles can complete a wheel graph. So far, all solutions have areas with 1 or 2 area radicals. Triangle Radicals has lists of triangles of a particular radical.

The following Heronian triangles all have the same perimeter. So they might be able to complete a wheel graph, or even complete an integral drawing. But likely not.

{{25, 25, 48}, {13, 40, 45}, {17, 40, 41}, {24, 37, 37}, {25, 34, 39}, {29, 29, 40}}

But there are many larger sets of equiperimeter triangles with the same radical. Someone could plow through them to see if there is a set to fill out a square.