# The Worst Case

Geometry Level 5

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 $$a + b$$?

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