The following suggestions are leading to a relationship in plane geometry attributed to Pappus. The history of mathematics cite on the link will give information about Pappas and some of his work.
Consider any triangle ABC.
Construct external parallelograms on sides AB and AC. These can be any parallelograms with one side the same length as the corresponding side of triangle ABC.
Extend the external parallels to intersect at point D and draw segment AD.
Use the direction and length of the segment AD to construct a parallelogram on the side BC.
Prove that the sum of the parallelogram areas on sides AB and AC equals the area of the parallelogram on the side BC. (Pappus Areas)
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Prove the Pythagorean Theorem using the Pappus Area theorem.
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