A property of Regular Octagons

Question
Prove that the area of a regular octagon is the product of its longest diagonal and its shortest diagonal

Solution

There’s an easier solution by cutting & pasting cleverly transforming the octagon into the area of a rectangle formed by cutting and pasting (where the length & width of this rectangle are the longest & shortest diagonals of this regular octagon) and the claim follows.

But, let’s look at a more “analytical way” though more complicated than the synthetic approach mentioned above.

The details are shown in the images (in order) beginning with the background facts/results needed prior & how to use it to formulate the solution accordingly: