HMMT November

Geometry Difficulty 4.5 AIME Find the answer

Let $A B C D$ be a quadrilateral with $A=(3,4), B=(9,-40), C=(-5,-12), D=(-7,24)$ . Let $P$ be a point in the plane (not necessarily inside the quadrilateral). Find the minimum possible value of $A P+B P+C P+D P$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

By the triangle inequality, $A P+C P \geq A C$ and $B P+D P \geq B D$ . So $P$ should be on $A C$ and $B D$ ; i.e. it should be the intersection of the two diagonals. Then $A P+B P+C P+D P=A C+B D$ , which is easily computed to be $16 \sqrt{17}+8 \sqrt{5}$ by the Pythagorean theorem. Note that we require the intersection of the diagonals to actually exist for this proof to work, but $A B C D$ is convex and this is not an issue.

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