# proof of when is a point inside a triangle

Let $\mathbf{u}\in {\mathbb{R}}^{2}$, $\mathbf{v}\in {\mathbb{R}}^{2}$
and $\mathrm{\U0001d7ce}\in {\mathbb{R}}^{2}$. Let’s consider the convex hull
of the set $T=\{\mathbf{u},\mathbf{v},\mathrm{\U0001d7ce}\}$.
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 $t\mathbf{u}+\left(1-t\right)\mathbf{v}\in C$,
$t\mathbf{u}\in C$ and $t\mathbf{v}\in C$ 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 $T$,

$${\mathrm{\Delta}}_{T}=coT=\{\mathbf{x}\in {\mathbb{R}}^{2}:\mathbf{x}=\lambda \mathbf{u}+\mu \mathbf{v}+(1-\lambda -\mu )\mathrm{\U0001d7ce},0\le \lambda ,\mu ,\le 1,0\le 1-\lambda -\mu \le 1\}$$ |

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 $0\le \lambda ,\mu ,\le 1$ 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 |