If I have a bounded set $F$ in $N$ dimensional space and another set $G$ where every element $g$ in $G$ has $h'g=c$ and also must exist in $F$. $H$ is a vector in the $N$ dimensional space and $c$ is any constant $1\times 1$ matrix (scalar). $h$ is a vector of appropriate dimension.
How can I prove every extreme point of $G$ lies on the boundary of $F$?
That is to say if $x$ and $y$ are extreme points in $F$ then $xλ + (1-λ)y = g$