Question 12
A given triangle $T$ decomposes into triangles $T_{1}, T_{2}, \ldots, T_{n}$ so that:
a) No pair of triangles $T_{i}$ has common interior points.
b) The union of all triangles $T_{i}$ is $T$.
c) Each segment that is side of some triangle $T_{i}$, or is side of another triangle $T_{j}$, or it is side of the triangle $T$. Let $s$ be the total number of sides (each counted only once, even if it is common to two triangles), and $v$ the total number of vertices (each counted only once, even if it is common to several triangles). Prove that if $n$ is odd, there are several decompositions of this class, and they all have the same number $v$ and the same number $s$. Express $v$ and $s$ in function of $n$. also show that if $n$ is par, there is no such decomposition.
a) No pair of triangles $T_{i}$ has common interior points.
b) The union of all triangles $T_{i}$ is $T$.
c) Each segment that is side of some triangle $T_{i}$, or is side of another triangle $T_{j}$, or it is side of the triangle $T$. Let $s$ be the total number of sides (each counted only once, even if it is common to two triangles), and $v$ the total number of vertices (each counted only once, even if it is common to several triangles). Prove that if $n$ is odd, there are several decompositions of this class, and they all have the same number $v$ and the same number $s$. Express $v$ and $s$ in function of $n$. also show that if $n$ is par, there is no such decomposition.