I stumbled upon this old question "Area of octagon constructed in a square" that involves finding the area of an octagon within a square as shown:
I found that the octagon is not regular. So, my question is this:
Prove that the octagon shown above is not regular, assuming the only length we are initially given is the side length of the square.
Note: The edges of the octagon are determined by the line joining the vertices of the square to midpoints of the square's sides.
So far, the easiest way I could think of was to use the cosine rule to get the largest angle in $\triangle ABC$ (below) and show that the total of the angles would not be equal to the expected sum of a regular octagon.
Are there any other ways of proving that this octagon is not regular?