Steiner's fundamental theorem - Why the $3$-gon is also equilateral? Related to the proposition $6$ (page $820$) of the article Sound of Symmetry, could anyone be able to tell me why the $3$-gon $P$ is equilateral (they use the Steiner's fundamental theorem (page $819$).) I think this is a trivial question, but I am stuck to answer it.

Steiner's fundamental theorem : Among all triangles with the same base and height, the isosceles has the smallest perimeter.

Thanks!

By definition, $P$ is a triangle that maximizes the ratio of area to perimeter squared.  This means that if $Q$ is any other triangle with the same area as $P$, the perimeter of $Q$ is greater than or equal to the perimeter of $P$.
Now suppose $P$ is not equilateral.  Choose one side of $P$ as a base such that the other two sides do not have equal length.  Let $Q$ be the isosceles triangle with that same base and the same height as $P$.  Then $Q$ has the same area as $P$, so it must have greater or equal perimeter.  But $P$ is not isosceles with respect to this base, so by Steiner's theorem $Q$ has smaller perimeter than $P$.*  This is a contradiction.
*As stated, Steiner's theorem may be interpreted to say the perimeter of $Q$ is less than or equal to the perimeter of $P$.  But if you look at the proof on page 819, it actually gives a strict inequality.
