Radon's theorem: Any $$d+2$$ points in $$R^d$$ can be partition into two disjoint sets whose convex hulls intersect.
Let these points be $$A=\{a_1,a_2,\ldots,a_{d+2}\}$$. Consider the point $$x=\frac{1}{d+2}\sum_{i=1}^{d+2}a_i$$. By definition $$x$$ is in convex hull of $$A$$.
By Caratheodory's theorem, $$x$$ can be represented as convex combination of at most $$d+1$$ points in $$A$$. We then have an equation whose left hand side and right hand side do not cancel each other because one side has $$d+2$$ non-zero terms and the other has no more than $$d+1$$. Moreover after rearrangement we get a point in $$R^d$$ which is the convex combination of two disjoint sets of $$A$$.