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For many geometric inequalities, the strategy of inscribing or circumscribing a figure can be useful (recall, for example, Application 1.1, Sections 1.3 and 1.6, and Theorem 2.3). Of all such inscribed or circumscribed figures, the circle plays a central role, and results in a variety of inequalities relating the radius of the circle to numbers associated with the given figure, such as side lengths, perimeter, area, etc.
The triangle is exceptional because every triangle possesses a circle passing through the vertices of the triangle, the circumcircle, whose center is the circumcenter of the triangle, and a circle inside the triangle and tangent to its three sides, the incircle, whose center is the incenter of the triangle.
Inscribing and circumscribing in the Elements of Euclid
Among the thirteen books of the Elements of Euclid (circa 300 BCE), there is one, Book IV, devoted to inscribing and circumscribing figures. This book starts with two basic definitions [Joyce]:
Definition 1. A rectilinear figure is said to be inscribed in a rectilinear figure when the respective angles of the inscribed figure lie on the respective sides of that in which it is inscribed.
Definition 2. Similarly a figure is said to be circumscribed about a figure when the respective sides of the circumscribed figure pass through the respective angles of that about which it is circumscribed.
After five complementary definitions, Euclid proves sixteen propositions. Euclid mainly considers problems of inscribing or circumscribing circles about triangles, squares, regular pentagons and hexagons, and regular 15-gons.
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