How many isosceles triangles do you need to make any polygon?

There is a two ears theorem which states for $n > 3$, any simple $n$-gon has at least two ears.

If one split a $n$-gon at an ear, we get a triangle and a $(n-1)$-gon.

For any triangle $ABC$ with $BC$ being the longest side.

  • If $ABC$ is a right triangle, the circumcenter $O$ coincides with midpoint of $BC$, we can decompose $ABC$ into two isosceles triangle $ABO$, $AOC$.

  • Otherwise, let $D$ be the foot on $BC$. We can split $ABC$ first into two right triangles $ABD$, $ADC$ and then into $4$ isosceles triangles.

In general, we can decompose any triangle into at most $4$ isosceles triangles.

By induction on $n$, we find we can split a $n$-gon into at most $4n$ isosceles triangles. This bound is probably not optimal but at least we know the decomposition is always possible.

Update

For general triangle, the bound $4$ is optimal.

If triangle $ABC$ is acute, we can decompose it into $3$ isosceles triangles: $AOB$, $BOC$ and $COA$. If $ABC$ is a right triangle, $2$ is enough. This leaves us with the case of obtuse scalene triangles.

A literature search indicate in $2004$, Kosztolányi, et al${}^{\color{blue}{[1]}}$ has studied the problem of decomposing obtuse scalene triangles into $3$ isosceles triangles.

With help of a computer, they found there are $23$ families of solutions. Let $\alpha > 90^\circ > \beta > \gamma$ be the angles of the trangle. In all these solutions, $\alpha, \beta$ are rational linear combinations of $180^\circ$ and $\gamma$. In particular, this implies triangle with angles $$(\alpha,\beta,\gamma) = \left( (5-\sqrt{2})\cdot 30^\circ, 30\sqrt{2}^\circ, 30^\circ \right)$$ cannot be decomposed into $3$ isosceles triangles. It is not hard to verify we cannot decompose this triangle into $2$ isosceles triangles. This means for general triangle, the bound $4$ is optimal.

Notes

  • $\color{blue}{[1]}$ - Kosztolányi, József & Kovács, Zoltán & Nagy, Erzsébet. (2004). Decomposition of triangles into isosceles triangles II. Complete solution of the problem by using a computer. Teaching Mathematics and Computer Science. 2. 275-300. 10.5485/TMCS.2004.0059.

    An online copy can be found here.

Tags:

Geometry

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