Pythagoras

1. Theorem of Pythagoras

Construct squares on the sides of a right angled triangle ABC. The square on the hypotenuse BC has area equal to the sum of the squares on the two other sides.

A classical proof results by dividing the big square into two parallelograms JKDC and JKEB through the altitude from the right angle A. Then showing that each of them has area equal to the square having a common vertex with the corresponding parallelogram.
In fact, parallelogram CDKJ has area twice the area of the triangle ACD. But this triangle is equal to GCB which also has area half the area of the square GCAF. Thus the areas of CDKJ and CGFA are equal. Similarly the areas of JKEB and ABHI are equal.

2. Generalization

Consider a fixed polygon p = MNO... and a right angled triangle ABC. Pick a side of the polygon, MN say, and glue it by similarity on the sides of the triangle. They result three similar polygons p_A (opposite to A), p_B,
p_C. The area of the polygon opposite to the right angle is the sum of areas of the two other polygons.

The proof is a consequence of the theorem of Pythagoras. In fact, denote the lengths of the sides of the triangle by a=BC, b=CA, c=AB. Assume that the polygon has area (e) and its side MN has length m. Then by the similarity:
area(p_A)/e = a²/m², area(p_B)/e = b²/m₂, area(p_C)/e = c²/m₂.
The proof results by substituting in the first equality a² = b² + c².

3. Remark

The previous property extens to shapes similar to a fixed one p having area e and having also a boundary containing some segment MN. Then one can glue by similarity the shape p to the sides of the right angled triangle and create three similar shapes p_A, p_B, p_C. Again the areas satisfy area(p_A) = area(p_B) + area(p_C).

Taking for shape p a half-circle and gluing it to the sides of the triangle we get as a result the well known theorem about the sum of the areas of the aqua colored lunes: the sum of their areas is equal to the area of the triangle.

There are many interesting discussions on Pythagora's theorem. Two of my favorits are: [YiuEGN, p.1] and [Bottema, p. 2].
See also the file SquaresDissection.html for another proof (of the innumerable existing) of this theorem.

Bibliography

[Bottema] O. Bottema Topics in Elementary Geometry Springer Verlag Heidelberg 2007
[YiuEGN] Yiu, P. Euclidean Geometry Notes (pdf). http://www.math.fau.edu/yiu/EuclideanGeometryNotes.pdf

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1. Theorem of Pythagoras

2. Generalization

3. Remark

See Also

Bibliography