Try this beautiful problem from Mathematical Circles book based on Triangle inequality.
Find a point inside a convex quadrilateral such that the sum of the distances from the point to the vertices is minimal .
Triangle Inequality
Inequality
Geometry
Mathematical Circles - Chapter 6 - Triangle Inequality - Problem 6
Mathematical Circles by Dmitri Fomin , Sergey Genkin , Llia Itenberg
Do you really need a hint ? You can start thinking about the Triangle Inequality...........
If you have already get the idea about the main concept for this sum then you can start the problem by taking a quadrilateral ABCD with diagonals that intersect at point 'o'.
The distance from 'o' to all vertices is equal.
Here is the diagram where OA + OB + OC + OD = AC + BD - which is sum of the diagonals. We can consider this as one case ...
As a last hint you have to take another point to o' to compare with first case
Now o' be another point inside the quadrilateral. If we use triangle inequality here we have ,
AO' + OC' > AC from Triangle AO'C
BO' + O'D > BD from Triangle BO'D
Hence AO' + O'C + BO' + O'D > AB + BD
Therefore its clear from this that the point that minimizes the sum of the distances is the point of intersection of diagonals.
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