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 |
---|---|
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 |