Math CentralQuandaries & Queries


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.
Prove that these segments have no common points.


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 $ABDC.$



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