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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