proof of when is a point inside a triangle


Let 𝐮∈ℝ2, 𝐯∈ℝ2 and 𝟎∈ℝ2. Let’s consider the convex hull of the set T={𝐮,𝐯,𝟎}. By definition, the convex hull of T, noted c⁢o⁢T, is the smallest convex set that contains T. Now, the triangleMathworldPlanetmath ΔT spanned by T is convex and contains T. Then c⁢o⁢T⊆ΔT. Now, every convex C set containing T must satisfy that t⁢𝐮+(1-t)⁢𝐯∈C, t⁢𝐮∈C and t⁢𝐯∈C for 0≤t≤1 (at least the convex combination of the points of T are contained in C). This means that the boundary of ΔT is contained in C. But then every convex combination of points of ∂⁡ΔT must also be contained in C, meaning that ΔT⊆C for every convex set containing T. In particular, ΔT⊆c⁢o⁢T.

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

ΔT=coT={𝐱∈ℝ2:𝐱=λ𝐮+μ𝐯+(1-λ-μ)𝟎,0≤λ,μ,≤1,0≤1-λ-μ≤1}

we conclude that 𝐱∈ℝ2 is in the triangle spanned by T if and only if 𝐱=λ⁢𝐮+μ⁢𝐯 with 0≤λ,μ,≤1 and 0≤λ+μ≤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