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A polygon is inscribed in a circle if all of the vertices of the polygon lie on the circle. Alternatively, a circle is inscribed in a polygon if the circle is tangent to every one of the polygon's edges. Not every polygon can be inscribed in a circle. Any polygon can be inscribed in at most one circle (because any two circles can intersect at no more than two points), and any polygon can have at most one circle inscribed in it.

Every (nondegenerate) triangle can be inscribed in a circle, however, and the resulting circle is called the circumcircle of the triangle.

A circle can also be inscribed in every nondegenerate triangle; this circle is called the incircle of the triangle.