All triangles are equilateral

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Well, I don't know if this is the reason, but it seems that it is possible to construct a triangle where the point M lies outside the triangle.

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Can you see what's wrong? Yes, I think so. Should I post it or would that ruin the fun for anybody? I'll just say that I think stand's comment above is in the right direction but not strong enough.

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In case AB is not equal to AC, M lies outside the triangle. If that's the case, |AC|=|RC|-|AR|, and not |AC|=|RC|+|AR|, as stated in the proof.

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