Friday, May 1, 2020

Proofs of the Pythagorean Theorem

There are hundreds of proofs of the Pythagorean Theorem. I've been testing my own limited knowledge of geometry trying to master as many as I can. Many of the proofs use similar techniques (recombining various triangular, square and rectangular shapes in various ways to show the desired result), so sometimes it is difficult to say if a proof is truly unique.

Of the many reconstruction methods, a popular technique is arranging multiple copies of a right-angle tri angle into a square shape then calculating the area of the resultant square in two ways. First, the squaring the length of a side. Second, calculating the areas of the components that make up the square. Setting these two areas equal, produces the desired result, showing:
C2 = A2 + B2

An example of this approach is given by Bhaskara's First Proof: Bhaskara's First Proof

The most concise proof, in my opinion, is Bhaskara's second proof: 

Start with a right-angle triangle, ABC. Let the lengths of the sides be a, b, and c.



Draw the altitude line, from point C to the opposite side at point D. Line CD is perpendicular to side AB. 

The three triangles, ABC, CBD and CAD are all similar. Let the distance from A to D be x and then from B to D is c - x.

Since the ratio of sides of similar triangles are equal,
 
a/c = (c-x)/a                              b/c = x/b

Simplifying,  

a2 = c2 – cx                                 b2 = cx

Adding these two equations together, results in (note, the term cx and -cx cancel),

 a2 + b2 = c2


The broken domino proof starts with a domino-shape tile that consists of two equal squares (each representing  c2). Three triangular pieces are broken from one of the squares and rearranged to form the squares representing a2 + b2. See proof 35 at Gary Zabel's page: Gary Zabel UMB

As a practical use of the broken tile proof, there are actual tiling patterns based on the Pythagorean Proof. Tiling with Pythagoras

A prior post outlines President James Garfield's proof of the Pythagorean Theorem. Garfield Proof

Wikipedia entry: Pythagorean Theorem

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