CTK Insights

03 Sep

Squares in Triangles

Here is a simple problem I found in a recent book from MAA (C. Alsina, R. B. Nelsen, Charming Proofs, p. 89.)

Geometric constructions of a square inscribed in a triangle are well known. There are two ways to inscribe a square into a right isosceles triangle:

Which square has the larger area?

The problem is simple and admits a pretty visual argument.


The side lines of the squares cut off the triangle smaller "leftover" triangles.

In both cases, the leftover triangles can be recombined to form squares equal to the drawn ones. In the second case a small triangle in the right corner is left unmatched, meaning that in that case the square is smaller.

We may clearly give a more accurate answer.

In the first case, the area of the square is exactly 1/2 that of the triangle. In the second case it is only 4/9.

A question begs to be asked: could any assertion of the above be generalized to triangles of a more generic shape?

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