We will triangulate (as in Problem 18) a convex polygon that has a convex polygonal hole cut out of its interior.
(a) Draw a large triangle and within this draw (and shade) a smaller triangle. Triangulate the resulting polygonal flat doughnut. How many triangles do you obtain?
(b) Draw a large pentagon and within this draw (and shade) a smaller triangle. Triangulate the resulting polygonal flat doughnut. How many triangles do you obtain?
(c) Draw a large hexagon and within this draw (and shade) a smaller square. Triangulate the resulting polygonal flat doughnut. How many triangles do you obtain?
(d) How many triangles does any triangulation of a polygonal flat doughnut have? Prove that your response is correct.
Problem 18
A triangulation of a convex polygon is a partition of that polygon into triangles that does not introduce any new vertices.
(a) Consider a triangle. How many triangles are in a triangulation of a triangle?
(b) There are different ways to triangulate a square. How many triangles does each have?
(c) How many triangles does any triangulation of an n-sided convex polygon have? Prove that your response is correct.