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!
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Let the base be $BC$. Consider a line parallel to the base at a distance equal to the height. We need to find the point $A$ on this line such that $AB+AC$ is minimum. If we consider a light source at $B$, the ray of light from $B$ that passes through $C$ travels the smallest distance. Hence by the principle of reflection, angle of incidence must be equal to the angle of reflection at $A$ and it immediately follows that $AB = AC$.
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.