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Triangulation, proof by string induction.

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Use strong induction to show that when a convex polygon P with consecutive vertices VI, V2, ... , Vn is triangulated into n - 2 triangles, the n - 2 triangles can be numbered 1, 2, ... , n - 2 so that Vi is a vertex of triangle i for i = 1,2, ... , n - 2.

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Solution Summary

Strong induction is used to prove that a convex polygon with n vertices can be triangulated into (n-2) triangles with the specified numbering of vertices.

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We'll triangulate the polygon like this: start with vertex number 1. Connect the two vertices adjacent to it (the last one and the second one) with a new edge. Number the resulting triangle 1. Now, look at the rest of the polygon. It's now a convex polygon with n-1 vertices, and the vertices are numbers 2, 3, 4, ... , n. Again, "remove" the second vertex by connecting the nth and the third ...

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