If all of the angles of a polygon are equal, and half of its sides are equal, then must the polygon be regular?

Question

If we have that all the angles in a 2d polygon are equal, and if we have that half the sides of the polygon are equal(for example, if the polygon has 10 sides, we know that 5 of them are equal), is it enough to prove that the polygon is regular? If so, how?

Answer 1

A rectangle serves as a counterexample; doesn't it?

Answer 2

this is not true

Here's a counterexample:

Answer 3

If the polygon has an even number of sides $n=2k$, than you need to have at least $n-1$ sides equal to be able to deduce that the polygon is regular. That's because regular $2k$ polygons have opposite sites in parallel, you can start with a regular $2k$ poloygon, choose one pair of parallel sides and then make those 2 side longer (or shorter) by the same amount at the same time. This means only $n-2$ sides is not enough.

The question becomes more interesting when you have a polygon with an odd number of sides. For example for $n=3$, you know it is a regular (equilateral) triangle already from knowing all it's angles are the same. If you are interested in that case, I suggest you make a new question.