Same side nooks
A concave polygon can be equilateral. For example, the concave octagon to the right is equilateral.
What is the least number of sides a concave equilateral polygon can have?
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1 solution
Moderator note:
As pointed out in the comments, another way to form an equilateral concave pentagon is to start with a regular pentagon and flip one of the points inward:
Suppose that it were possible to have a concave equilateral quadrilateral.
Consider the triangles △ A B C and △ A D C . If A B C D is equilateral, then A B = B C = A D = D C . In addition, the triangles share the side A C . The triangles are congruent by SSS congruence, but this implies they are coincident. This is a contradiction. Thus, a concave equilateral quadrilateral is impossible.
A concave equilateral pentagon is possible, although they typically look kind of weird.
This particular concave equilateral pentagon was formed by taking a rhombus whose smallest angle is less than 60 ∘ , and then placing an equilateral triangle on top of it. However, there are other ways of forming a concave equilateral pentagon.