The area of any quadrilateral is equal to half the product of the diagonals multiplied by the sine of the angle between them. As shown in the dynamic sketch below the area ABCD = ½AC x BD x sin ∠DEA.
It's surprising though that this simple formula appears to be completely absent from high school textbooks.
The reader/viewer is requested to investigate the validity off the result by dragging any of the red vertices A, B, C or D.
1) Can you explain why (prove that) the above result is true?
2) Drag the vertices so that ABCD becomes concave or crossed. Can you also explain why (prove) that the result remains true in these cases?
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Michael de Villiers, created 29 January 2020.