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Levi Pearson 2014-01-25 00:36:19 -07:00
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@ -72,3 +72,23 @@ polyhedrons can be tetrahedralized!
: Three consecutive vertices $a, b, c$ form an *ear* of a polygon if : Three consecutive vertices $a, b, c$ form an *ear* of a polygon if
$a c$ is a diagonal of the polygon. The vertex $b$ is called the ear $a c$ is a diagonal of the polygon. The vertex $b$ is called the ear
*tip*. *tip*.
**Corollary 1.9**
: Every polygon with more than three vertices has at least two ears.
A vertex of a polygon is *reflex* if its angle is greater than $\pi$
and *convex* if its angle is less than or equal to $\pi$. It is *flat*
if its angle is exactly $\pi$ and *strictly convex* if its angle is
strictly less than $\pi$.
A polygon $P$ is a *convex polygon* if all of its vertices are
strictly convex.
**Lemma 1.18**
: A diagonal exists between any two nonadjacent vertices of a polygon
$P$ if and only if $P$ is a convex polygon.
**Theorem 1.19**
: The number of triangulations of a convex polygon with $n + 2$
vertices is the Catalan number $$ C_n = \frac{1, n + 1}{n + 1 \choose
n} $$