## Maximum area of quadrilateral problem

If *E* is a fixed point inside quadrilateral *ABCD* with the distances *EA*, *EB*, *EC* and *ED* fixed, when will *ABCD* have its the maximum possible area? Explore the problem experimentally by using the dynamic sketch below and dragging the vertices.

Drag any of the red vertices *A*, *B*, *C* or *D* in the sketch below.

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Maximum area of quadrilateral problem

What conjecture did you form? Can you explain (prove) why your conjecture is correct?

This problem was used as Question 20 in Round 1 of the 2012 Senior South African Mathematics Olympiad. The questions and solutions are available at *Question Papers and Solutions.*

Now consider the following related problem. When will quadrilateral *ABCD* have its the maximum possible area if its diagonals *AC* and *BD* are of fixed length? Explore the problem experimentally by using the dynamic sketch below and dragging the vertices.

Drag any of the red vertices *A* or *D* in the sketch below.

By dragging the respective midpoints *I* and *H* of the diagonals *AC* and *BD*

you can also change the position where the diagonals intersect without changing the length of the diagonals.

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Maximum area of quadrilateral problem 2

What conjecture did you form? Can the maximum area be achieved with quadrilaterals of different shape? Does the quadrilateral with maximum area have to be convex? Can you explain (prove) why your conjecture is correct? Check your logical explanation (proof) at *Solution.*

This page uses **JavaSketchpad**,
a World-Wide-Web component of *The Geometer's Sketchpad.*
Copyright © 1990-2011 by KCP Technologies, Inc. Licensed only for
non-commercial use.

Modified by Michael de Villiers, 10 June 2012.