



 
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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