A circle is drawn that intersects all three sides of triangle PQR as shown below. Prove that if AB = CD = EF, then the center of the circle is the

A circle is drawn that intersects all three sides of triangle PQR as shown below. Prove that if AB = CD = EF, then the center of the circle is the incenter of triangle PQR.

Take O as the center of the circle

Draw OG, OH, OI   perpendicular  to  AB, CD  and EF  respectively

And, by Euclid.....equal chords are equal distances from the center of  a circle

So GO = HO and OQ is common

So triangles QOG  and QOH are congruent right triangles by HL

So < GQO  = < HQO

By similar reasoning, triangles ROH and ROI  are congruent right triangles by HL

So < HRO  = < IRO

And once more, triangles POI and POG  are congruent right triangles by HL

So < IPO  = < GPO

Therefore......since QO, RO and PO  are angle bisectors of triangle PQR that meet at O, then O is the triangle incenter.....but O is also the center of the circle