A Little Geometry

I’ve just finished the book “Euclid’s Window” by Leonard Mlodinow, and really enjoyed it. The book describe the history of geometry from Euclid, Descartes, Gauss, and Einsten. During his coverage of Euclid he presents a simple proof of the Pythagorean Theorem that really resonated with me. I don’t recall ever seeing a proof of it, or at least no memorable proof. This one uses a minimum of jargon and formality…you just draw the picture, discuss it for a bit, and you see it!

You start with a right triangle, like:

and you make two constructions, from a square with sides a+b. The first construction looks like:

which, by eye, you can see that the total area of the square is the area of 4 triangles (just like our original) plus the area of the inner square, which is c*c (which reminds me that I have to figure out how to do superscripts and subscripts in this blog. 🙂 )

The second construction is nearly the same as the first, and looks like:

which, again by eye (with a little shading to make it a bit more obvious), the total area of the square is the area of 4 triangles (just like our original) plus the area of the two inner squares, which are a*a and b*b. Therefore:


for any triangle for which you can make this construction, which are right triangles.

Really neat!


About brianblais

I am a professor of Science and Technology at Bryant University in Smithfield, RI, and a research professor in the Institute for Brain and Neural Systems, Brown University. My research is in computational neuroscience and statistics. I teach physics, meteorology, astonomy, theoretical neuroscience, systems dynamics, artificial intelligence and robotics. My book, "Theory of Cortical Plasticity" (World Scientific, 2004), details a theory of learning and memory in the cortex, and presents the consequences and predictions of the theory. I am an avid python enthusiast, and a Bayesian (a la E. T. Jaynes), and love music.
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