While constructing an equilateral triangle or a regular hexagon inscribed in a circle, you may have noticed that several smaller equilateral triangles are formed, like pqr shown in the figure below. explain why pqr is an equilateral triangle.

ΔPQR is an equilateral triangle because its sides are equal. 

In a regular hexagon, the r1 or radius of the incircle is equal to the side length of the hexagon.

Segment RQ is the side length of a hexagon that will be inscribed in circle Q. Segment RP is the side length of a hexagon that will be inscribed in circle P. Both circles share segment PQ which is its radius. 

Since radius of the incircle is equal to the side length of the hexagon, all sides of the triangle are equal, thus, creating an equilateral triangle.

Attached is a picture of a regular hexagon, its parts and formulas. It is simply to help you understand its relation to the circle and triangle.