*The
Triangulation of Any Polygonal Region*
in the Plane

*The*** triangulation of any polygonal region****
in the plane is a key element in a proof of the equidecomposable
polygon
theorem.**

Proposition:** ***Any region in the plane
bounded by a closed polygon can be decomposed into the union of a
finite number of closed triangular regions which intersect only on the
boundaries.*

The proof of this proposition examines a more careful characterization
of the polygonal regions being considered. The key idea of the proof
goes
by induction on the number n = the number vertices = the number of
sides
in the polygon, as follows:

__Proof__: Consider ** n = the number vertices =
the number of sides
in the polygon**

**When n = 3 the result is trivial.**

__Suppose n> 3__** and that for any polygon with k vertices/
sides, where
k<n, the polygon can be triangulated. [An induction hypothesis.]**

**Now proceed to consider the vertices, v1, v2, ..., vn ordered
so that**__ vi is adjacent to v(i+1) and vn is adjacent to v1__.

**Take a ray from v1 and rotate it from v1v2 so that it intersects
the inside of the polygon. Continue to rotate until it meets another
vertex.**

Case 1.** This vertex is v3. Then consider
the polygonal region Q1
= v1v3...vn which has n-1 vertices. By induction Q1 can be
triangulated,
so the original polygon is triangulated using the triangulation of Q1
and
the triangle v1v2v3.**
**Case 2. The vertex is v(n-1). Then consider
the polygonal region Q2
= v1v2v3...vn-1 which has n-1 vertices. By induction Q2 can be
triangulated,
so the original polygon is triangulated using the triangulation of Q2
and
the triangle v1vnv(n-1).**

**Case 3. The vertex is vk with k different
from 3 or n. Then consider
the polygonal regions Q3 = v1v2...vk which has k vertices (k<n) and
Q4 = v1vkv(k+1)...vn which has n-(k-2)<n vertices. By induction Q3
and
Q4 can be triangulated, so the original polygon is triangulated using
the
triangulations of Q3 and Q4.**

**This completes the argument that allows the use of induction to
finish the proof of the proposition.**

**For more discussion of proofs of this proposition see Triangulations
and
arrangements, Two lectures by Godfried Toussaint, transcribed by
Laura Anderson and Peter Yamamoto.**