Answer
each of these two polygons has two triangles that have two sides that border their exterior, and
in each case only one of these triangles can fail to be a triangle
that has two sides that border the exterior of the original
polygon
Work Step by Step
--Let P(n) be the statement that
-if a simple polygon with n sides is triangulated, then at least two
of the triangles in the triangulation have two sides that border
the exterior of the polygon.
-We will prove ∀n ≥ 4 P(n).
The statement is clearly true for n = 4,
-because there is only one diagonal, leaving two triangles with the desired property.
-Fix k ≥ 4 and assume that P(j) is true for all j with 4 ≤ j ≤ k.
Consider a polygon with k + 1 sides, and some triangulation
of it. Pick one of the diagonals in this triangulation.
-First suppose that this diagonal divides the polygon into one triangle
and one polygon with k sides.
-Then the triangle has two sides that border the exterior.
- Furthermore,
--the k-gon has, by the inductive hypothesis,
-two triangles that have two sides that
border the exterior of that k-gon, and only one of these triangles
can fail to be a triangle that has two sides that border the
exterior of the original polygon.
-The only other case is that this diagonal divides the polygon into two polygons with j sides and k + 3 − j sides for some j with 4 ≤ j ≤ k − 1.
--By the inductive hypothesis,
- each of these two polygons has two triangles that have two sides that border their exterior, and
in each case only one of these triangles can fail to be a triangle
that has two sides that border the exterior of the original
polygon
