Equivalent formulation of convex functions

Question

I came across an equivalent formulation for convexity of a function and that is a function f is convex if and only only $f(y+$$\beta$$(y-x))$ $\geq$ $f(y)$ + $\beta$$.$$(f(y)-f(x))$ where $x,y$ $\in$ Domain($f)$ and $\beta$ is such that $y+$$\beta$$(y-x)$ $\in$ Domain($f$). $\beta$ is non negative. What can be a geometrically convincing arguement to believe in this inequality?