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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

But try the problem first...

Source

Suggested Reading

Mathematical Circles - Chapter 6 - Triangle Inequality - Problem 6

Mathematical Circles by Dmitri Fomin , Sergey Genkin , Llia Itenberg

First Hint

*Do you really need a hint ? You can start thinking about the Triangle Inequality...........*

Second Hint

**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 ...*

Final Step

**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.*

- https://www.cheenta.com/triangle-inequality-theorem-explanation/
- https://www.youtube.com/watch?v=gF9EbKk7gT4&t=28s

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