Median Points in the Plane

For points $P_1,\dots,P_n$ in a plane $\alpha$, call a point $P$ (in $\alpha$) a median point if the sum of distances from $P$ to $P_1,\dots,P_n$ is minimal. (For example, every point of segment $AB$ is a median point of the endpoints $A,B$.) Determine, with proof, which of the following are true.

1. If $A,B,C$ are collinear with $C$ on segment $AB$, then $C$ is a median point of $A,B,C$. 2. The midpoint of the hypotenuse of a right triangle is a median point of the three vertices. 3. If $A,B,C,D$ are collinear, their median point exists and is unique. 4. The intersection of the diagonals of a trapezoid is its unique median point (of the four vertices).