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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