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.)

(In reply to Haha. by broll)

Construct line AB, with A and B as points on it.
Construct square ABCD, with C,D beneath AB.
Construct point 'handle' on AB between A and B. (as handle, for later use)
Construct circle O1 radius A'handle' on A.
Construct circle O2 radius A'handle' on B.
Let O1 and O2 intersect at C1 above AB.
Construct isoceles triangle ABC1

First Square: Construct rays C1C, C1D. The Intersections of these rays and AB are F,G.
Construct Square FGHI. H and I are on BC1 and AC1 respectively. FGHI has side length k.

At C1, construct circle O3, radius k. This circle intersects AC1 at J, and BC1 at N.
Construct squares C1JLM and C1NPQ on the insides of AC1 and BC1.

All 3 squares are now the same size.
Move 'handle' along AB until it reaches G. Hey presto.

Measure angle AC1B. It is 101.7359477... degrees. The ratio of the sides to the base is 1.551387524... By elimination, it seems to be the only other solution.

   

Edited on February 27, 2016, 10:35 pm

Posted by broll on 2016-02-27 07:47:04