TITLE: Dissect square into triangles of same perimeter
QUESTION [7 upvotes]: 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?

REPLY [3 votes]: 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.