When I was young and helping my Pa make the back porch he had me measure one side of the frame 4 feet then the other side 3 feet. He said that it was "square" when the distance between my two marks measured 5. 3 squared + 4 squared = 5 squared. That was the first time I used the Pythagorean Formula.
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The Pythagorean Theorem allows the mathematician to determine the value of the hypotenuse. The converse of the Pythagorean Theorem manipulates the formula so that the mathematician can use the values to determine that if the triangle is a right triangle.
Pick's theorem can't use for non-convex polygons. It needs at least 3 terms to define an area of a polygon.
You use the pythagorean theorem.
Germain's Theorem is about Vibrating Elastic Plates.