Every triangle has a square which is the maximum size square which can be inscribed in the triangle. For most triangles there is only one way to do so. For an equilateral triangle there are three ways to inscribe the square - one for each side.

The equilateral is not the only triangle with that property; there is one other triangle whose maximum inscribed square can be placed on all three sides. Determine the dimensions of that triangle! (Assume the shortest edge is 1 unit.)

This is what I would call an ingeniously worded problem.

When we talk about inscribed squares, normally what is meant is a square with all four corners touching the sides of the triangle. A little tinkering with Geogebra shows there are no acute candidates. But then when the largest angle exceeds 90 degrees, two of the squares become 'undefined.'

But so what. Start with an obtuse triangle, and construct the squares; two of them stick outside the triangle a bit, but never mind. But in any case there is no example where all 3 squares are the same, between 90 degrees and 135 degrees.

Hang on. Since we already sacrificed a corner when searching, what about the case with 3 corners on the triangle, and the other a little inside? It turns out there is such an isosceles triangle with the large angle around 102 degrees.

If I'm able to produce a more exact construction, I'll give a more precise answer.

Posted by broll on 2016-02-27 06:36:01