proof of when is a point inside a triangle
Let , and . Let’s consider the convex hull of the set . By definition, the convex hull of , noted , is the smallest convex set that contains . Now, the triangle spanned by is convex and contains . Then . Now, every convex set containing must satisfy that , and for (at least the convex combination of the points of are contained in ). This means that the boundary of is contained in . But then every convex combination of points of must also be contained in , meaning that for every convex set containing . In particular, .
Since the convex hull is exactly the set containing all convex combinations of points of ,
we conclude that is in the triangle spanned by if and only if with and .
|Title||proof of when is a point inside a triangle|
|Date of creation||2013-03-22 17:57:08|
|Last modified on||2013-03-22 17:57:08|
|Last modified by||joen235 (18354)|