I recently came across a proof for the Pythagorean Theorem
that is simple and contains a minimum amount of equations. It may be one of the
most direct and understandable proofs of the many that exist. A nice animation of this proof is found at MathAdam.
Start with your standard right-angle triangle of sides A, B
and C with C being the hypotenuse.
Draw a line perpendicular to C to the opposite vertex.
Now, in addition to the original triangle, there are two
additional right-angle triangles formed. One has the side A as its hypotenuse (well
name triangle A) and the other has side B has its hypotenuse (triangle B).
The original triangle with sides ABC will now be called triangle C. The
three triangles, A, B, and C are all similar A ~ B ~ C.
(This can be established by the fact all three has the same interior angles.)
Another key fact that we can see is the area of A + B = area of C
The next step is to flip over each of the three triangles.
The next step is to the draw out
squares on each of the three sides.
The relative sizes of the squares
have the same relative ratios as the triangles A:B:C.
Since the areas of triangles has
the relationship: A + B = C, then the same applies for the squares,
therefore A2 + B2 = C2.
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