In many elementary (and not-so-elementary) Euclidean geometry texts, a (simple) polygon is said to be tangential if it is convex and has an inscribed circle (i.e., a circle that intersects and is tangent to each side of the polygon). The assumption of convexity is not needed: I've come up with a rather laborious proof that every polygon with an inscribed circle is convex. But I'd like to find either a simple elementary proof or a reference to a proof in the literature. (By "elementary," I mean using only standard facts of axiomatic Euclidean geometry.)
Does anyone know of a reference for a proof of this fact (elementary or not)? Or can anyone think of a straightforward elementary proof? You can use any definition of "convex polygon" that you like, but the easiest one to work with is that for each edge, the vertices not on that edge lie on one side of the line through that edge.
(Interestingly, the corresponding fact for circumscribed circles--i.e., that every polygon with a circumscribed circle is convex--is quite easy to prove: If P has a circumscribed circle, any two nonadjacent sides of P are non-intersecting chords of the circle; it is easy to show that both endpoints of each chord lie on the same side of the line through the other, and from there it is an easy matter to prove that P is convex.)