The Worst Case

3 friends have built their house on a flat piece of farmland. They want to select a drop-off point where they can leave messages for each other. Let their houses be located at $(x_1, y_1), (x_2, y_2),$ and $(x_3, y_3)$ respectively.

They want to minimize the total distance to walk to the drop-off point. The naive positioning would be to place it at the coordinate wise median, defined by coordinates $x_m=\text{median}(x_1,x_2,x_3) , y_m=\text{median}( y_1 , y_2 , y_3)$ , which we denote as $G$ . Let, the location that minimizes the sum of euclidean distances from the drop-off point to the houses be denoted as $G^*$ .

Let $sc ( A)$ denote the sum of euclidean distances from the point $A$ to each of these 3 houses. Over all possible placements of the houses, the maximium value of $\frac{ sc(G)} { sc(G^*) }$ can be written as $\frac{ \sqrt{a} } { b }$ , where $a$ and $b$ are relatively prime positive integers. What is $a + b$ ?