by Oldboy
Last Updated April 25, 2018 20:20 PM

If a plane shape $\phi$ has three different axes of symmetry (all belonging to the shape plane), they all interesect at the same point.

This is fairly obvious but I could not find an elementary proof.

It's relatively easy to prove that the plane shape cannot have two parallel axis of symmetry.

If the three axis of symmetry $p,q,r$ intersect without a common point, we can use the fact that composition of three different mirror reflections $S_p\cdot S_q\cdot S_r$ is a gliding reflection. And a gliding relfection cannot transform plane shape $\phi$ onto itself (to prove that I had to use the fact that a gliding reflection applied twice is a translation and also the fact that a plane shape has finite dimensions).

But I feel like I'm missing something pretty basic here... A simple proof based on elementary geometry.

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