Question 4
For three points $A,B,C$ in the plane we define $m(ABC)$ to be the smallest length of the three altitudes of the triangle $ABC$, where in the case of $A,B,C$ collinear, $m(ABC)=0$. Let $A,B,C$ be given points in the plane. Prove that for any point $X$ in the plane,
$$m(ABC)\le m(ABX)+m(AXC)+m(XBC).$$
$$m(ABC)\le m(ABX)+m(AXC)+m(XBC).$$