# proof of when is a point inside a triangle

Let $\mathbf{u}\in\mathbb{R}^{2}$, $\mathbf{v}\in\mathbb{R}^{2}$ and $\mathbf{0}\in\mathbb{R}^{2}$. Let’s consider the convex hull of the set $T=\left\{\mathbf{u},\mathbf{v},\mathbf{0}\right\}$. By definition, the convex hull of $T$, noted $coT$, is the smallest convex set that contains $T$. Now, the triangle $\Delta_{T}$ spanned by $T$ is convex and contains $T$. Then $coT\subseteq\Delta_{T}$. Now, every convex $C$ set containing $T$ must satisfy that $t\mathbf{u}+\left(1-t\right)\mathbf{v}\in C$, $t\mathbf{u}\in C$ and $t\mathbf{v}\in C$ for $0\leq t\leq 1$ (at least the convex combination of the points of $T$ are contained in $C$). This means that the boundary of $\Delta_{T}$ is contained in $C$. But then every convex combination of points of $\partial\Delta_{T}$ must also be contained in $C$, meaning that $\Delta_{T}\subseteq C$ for every convex set containing $T$. In particular, $\Delta_{T}\subseteq coT$.

Since the convex hull is exactly the set containing all convex combinations of points of $T$,

 $\Delta_{T}=coT=\left\{\mathbf{x}\in\mathbb{R}^{2}:\mathbf{x}=\lambda\mathbf{u}% +\mu\mathbf{v}+\left(1-\lambda-\mu\right)\mathbf{0},0\leq\lambda,\mu,\leq 1,0% \leq 1-\lambda-\mu\leq 1\right\}$

we conclude that $\mathbf{x}\in\mathbb{R}^{2}$ is in the triangle spanned by $T$ if and only if $\mathbf{x}=\lambda\mathbf{u}+\mu\mathbf{v}$ with $0\leq\lambda,\mu,\leq 1$ and $0\leq\lambda+\mu\leq 1$.

Title proof of when is a point inside a triangle ProofOfWhenIsAPointInsideATriangle 2013-03-22 17:57:08 2013-03-22 17:57:08 joen235 (18354) joen235 (18354) 4 joen235 (18354) Proof msc 51-00