Pythagorean Theorem – Cool Proof

I was taken with a “Proof Without Words” of Ptolemy’s Theorem by William Derrick and James Hirstein in an old(Nov 2012) edition of The College Mathematics Journal.  I had this vision of triangles expanding and rotating into place to form a parallelogram.  What a neat proof! I wanted to do one.  Since Ptolemy’s Theorem, “In an inscribed quadrilateral, the product of the diagonals is equal to the sum of the products of opposite sides” is a generalization of the Pythagorean Theorem why not try to prove Pythagorean Theorem using Derrick and Hirstein’s technique. Here is my attempt.

Pythagorean Theorem Proof - Part I

Pythagorean Theorem Proof – Part I

But the Pythagorean Theorem works both ways.  If we have a right triangle labeled in the standard way then a^2+b^2=c^2 and if a^2+b^2=c^2 then we have a right triangle.  Here is a proof of the second part.

Pythagorean Theorem Proof - Part II

Pythagorean Theorem Proof – Part II

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About jrh794

I am a sixty-five year old math instructor at Southern Oregon University. I taught at the College of the Siskiyous in Weed California for twenty-six years. Prior to that I worked as a computer programmer, carpenter and in various other jobs. I graduated from Rice University in 1967 and have a MS in Operations Research from Stanford. In the past I have hand-built a stone house and taken long solo bicycle tours. Now I ride my mountain bike and play golf for recreation.
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