Label the intersection of the circle C. Construct AC and BC.

Now AC and AB are equal because they are radii of the same circle.

Similarly BC and AB are equal.

Therefore, ...

Begin with the two parallelograms ABCD and EBCF between two parallels.

The first step of the proof is to establish that the triangles ABE and DCH are congruent.

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Construct a circle of radius AB with center at A.

Next, construct a circle of radius AB with center at B.

 

Therefore, AC and BC are equal, because both are equal to AB.

It follows that the sides of the triangle ABC are all equal. So, triangle ABC is equilateral, which was to be proved.

QED

Since triangles ABE and DCH are congruent and hence equal, the parallelograms ABCE and EBCH must be equal.

QED