Are all equiangular odd polygons also equilateral?

In standard Euclidean geometry, are all equiangular polygons with an odd number of sides also equilateral?

It is easy to prove that all equiangular triangles are also equilateral using basic trogonometric rules.

On the other hand, it is easy to conceive of an equiangular quadrilateral that is not equilateral, i.e. a rectangle.

Extending this further, I can easily conceive of an equiangular hexagon that is not equilateral, but I haven't been able to visualize an equiangular pentagon that is also equilateral: Is this true that all equiangular polygons with an odd number of sides also equilateral? If so, is there a straightforward way to prove it? If not, is there a counterexample, an equiangular polygon with an odd number of sides that is not equilateral?

• As a note, all cyclic equiangular odd polygons are equilateral. – user706071 Jun 21 '14 at 11:20
• And, all cyclic convex equilateral polygons are equiangular. – Oscar Lanzi Jun 23 at 19:50