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tangential convex polygon, touchpoints create polygon with same angles, regular?

Source: 2007 Sharygin Geometry Olympiad Correspondence Round P7

April 25, 2019
convex polygonInscribed circleregular polygongeometry

Problem Statement

A convex polygon is circumscribed around a circle. Points of contact of its sides with the circle form a polygon with the same set of angles (the order of angles may differ). Is it true that the polygon is regular?