proof of when is a point inside a triangle

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

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

$${\mathrm{\Delta }}_T=coT=\{𝐱\in {ℝ}^2:𝐱=\lambda 𝐮+\mu 𝐯+(1-\lambda -\mu )\mathrm{𝟎},0\le \lambda ,\mu ,\le 1,0\le 1-\lambda -\mu \le 1\}$$

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

Title	proof of when is a point inside a triangle
Canonical name	ProofOfWhenIsAPointInsideATriangle
Date of creation	2013-03-22 17:57:08
Last modified on	2013-03-22 17:57:08
Owner	joen235 (18354)
Last modified by	joen235 (18354)
Numerical id	4
Author	joen235 (18354)
Entry type	Proof
Classification	msc 51-00