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| Math Central | Quandaries & Queries |
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Question from Tim, a student: For this problem I actually have tried to visualise the image in my head many times. This question makes my head spin. Four points lie in a plane. They are partitioned into two pairs so that the sum of the lengths of the segments joining the points of each pair has the minimal possible value. |
Tim,
You might try proving that if the segments $AC$ and $BD$ intersect in a single point $P$, then $AC + BD > AD + BC.$ (Just use the triangle inequality on the triangles $PBC$ and $PDA,$ and observe that $BP+PD = BD$ and similarly for $AC.$)
It remains to look at when $A,B,C,D$ lie along a line and segment $AC$ overlaps segment $BD.$ Just draw the picture and observe that you can devise a smaller sum than AC + BD when the points are in the order $ABCD$ and also $ABDC.$
Chris
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