## 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.

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.

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.