Given is a triangle on points x,y,z in the plane. This triangle has two points a and b on different sides. I would like to show that the following inequality has to hold:
$\max \{d(b,x), d(b,y), d(b,z)\} + \max \{d(a,x), d(a,y), d(a,z)\} - d(b,a) \geq \min \{d(x,y), d(x,z), d(y,z)\}$
where d(u,v) denotes the euclidean distance between u and v. I actually expect the above statement to be true even if a and b are two arbitrary points outside of the triangle.
Does anybody have an idea how to approach this?