MathDB

Let

\mathcal{P}

be a convex polygon and

<span class='latex-bold'>T</span>

be a triangle with vertices among the vertices of

\mathcal{P}

. By removing

<span class='latex-bold'>T</span>

from

\mathcal{P}

, we end up with

0, 1, 2,

3

smaller polygons (possibly with shared vertices) which we call the effect of

<span class='latex-bold'>T</span>

. A triangulation of

P

is a way of dissecting it into some triangles using some non-intersecting diagonals. We call a triangulation of

\mathcal{P}

\underline{\text{beautiful}}

, if for each of its triangles, the effect of this triangle contains exactly one polygon with an odd number of vertices. Prove that a triangulation of

\mathcal{P}

is beautiful if and only if we can remove some of its diagonals and end up with all regions as quadrilaterals.

Problem Statement