Prove that a regular pentagon should be a cyclic polygon.

As follows:

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Actually, not only a regular pentagon but also all regular polygons are cyclic polygons.

Let us prove this concyclic property using the definition of regular polygons and circles without using the values of the angles and other derived properties of circles.

Definition: A regular polygon is a polygon of which all angles are congruent and all sides have the same length.

Let A, B, C, ..., N be the angles of the regular polygon. First, draw the angle bisector of A and B. Let the two bisectors intersect at O.

Since ∠OAB = ∠OBA , OA = OB .

Also, ∠OBA = ∠OBC (OB is the angle bisector)
and AB = BC (the sides of a regular polygon have same length)

Therefore, triangle OBA and triangle OBC are congruent (SAS).
Hence OB = OC. Similarly, we can prove OA = OB = OC = OD = ... = ON

We draw a circle centred at O with radius OA.
By the definition of circle, all the angles of the regular polygon A, B, ..., N lie on the drawn cirle.