Main Index Geometry Plane Geometry
  Subject Index
comment on the page

Pythagorean Theorem

The Pythagorean theorem is one of the most fundamental result of all euclidean geometry ¹ :

Pythagoras’ theorem: If is the right-angle triangle with legs , and  hypotenuse then .

In words: In a right-angled triangle the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle. ²

The following simple and short proof goes back to Bhaskara II (1114 - 1185) (It can be also found in Leonardo’s Practica geometriae (1220) and it was independently also discovered by Wallis [1] ) The right-angle triangles , and are similar (Euclid’.s Elements Book VI Prop. 8). Since the corresponding sides of similar triangles are proportional, , that is . ³ Similarly or . Substituting for we get .

The above proof is very simple and it is very probable that it was known to the old Babylonian. For instance, there is an interesting Babylonian Tablet known as Plimpton 322 which documents that a formula for the generation of triples of positive integers satisfying pythagorean equation was known in Babylon at least 1800 BCE (see also [2] ) .

Euclid’s proof (Elements Proposition 1.47) is purely geometric: The side-angle-side theorem implies that the triangle is congruent to . On the other hand, the triangle has half of the area as the square , for both have the same base and height. From similar reasons the area of is half of the area of the rectangle . Consequently the area of the square equals the area of the rectangle . Parallel reasoning can be used to show that the area of the smaller square equals the area of the smaller rectangle.

There are many other  known proofs of this theorem , most impressive are visual ones  by dissection which reassemble two small squares into the larger one . The following adaptation of the original tesselation proof due to Annairizi of Arabia (ca. 900 AD) can be found in [1] , p.22.

The following simple proof without words for isosceles right-angle triangle was found by the German philosopher A.Schopenhauer (1788-1860).  That the proof covers only a very special case of an isosceles righ-angle triangle did not discomfort him: “There must be a modification  of this schematic proof also for a non-isosceles triangle ...“ ⁴ (Die Welt als Wille und Vorstellung (The World as Will and Representation also known under the title The World as Will and Idea), Vol. 1, Chapter 15).

On another place he writes: “I am convinced that every, even the most intricate resut ... is reducible to such a simple illustration.” ⁵ (Über die vierfache Wurzel des Satzes vom zureichenden Grunde  (On the Fourfold Root of the Principle of Sufficient Reason), § 39).

From the planar generalizations of Pythagorean theorem let us mention two:

The law of cosine: For a triangle with sides and the angle opposite the side c, one has  .

Pappus’ Theorem  (Pappus [3] , Book IV) ↑ : Let be any triangle and , and are arbitrary parallelograms drawn on , and , resp. Let be the intersection of the lines and . Let and be segments equal and parallel to . Then .

The proof is similar to that of Pythagorean theorem as found in Euclid’s Elements. Since , and are parallelograms and . Similarly for . Similarly .

Euclid’s Proposition 31 (Book VI): In right-angled triangles the figure on the side opposite the right angle equals the sum of the similar ⁶  and similarly described figures on the sides containing the right angle.

This generalization says that instead of squares one can take semicircles, or generally if one erects similar figures on the sides of a right-angle triangle, then the sum of the areas of the two smaller ones equals the area of the larger one.

For polygons this result follows from the ratio of areas of similar polygons equals the ration of corresponding sides. When we draw the semicircle in last picture not under but above the hypotenuse then there follows the theorem involving lunes of Hippocrates : The sum of areas of lunes of Hippocrates overs the legs equals the area of the triangle. This result appears for the first time in Elemens de Geometrie written by Jesuit Pardies in 1671, who comments it as known, but the result does not appear in the work of Hippocrates. Hippocrates only quadratured the lunes. [1]

The converse of the Pythagorean theorem is also true, and it already appears in Euclid’s Elements (Elements Proposition 1.48): If are three positive numbers such that then there exists a triangle with sides and every such triangle has a right angle between the sides of length and (or if in a triangle the square on one of the sides equals the sum of the squares on the remaining two   sides of the triangle, then the angle contained by the remaining two sides of the triangle is right).

The following ingenious generalization of Pythagorean theorem goes back (at least) to Ptolemy   (~ 150 C.E.) and can be found in his Almagest (Book I, Ch. 10) :

Ptolemy’s Theorem: Let a quadrilateral be inscribed in a circle. Then the sum of the products of the two pairs of opposite sides equals the product of its two diagonals, that is .

Instead of the right-angle triangle consider the rectangle . Then the original statement of the Pythagorean theorem can be rephrased in terms of lengths of opposite sides and the diagonals. But this observation is to simple. Due to the similarity of triangles and we see that the angles and are equal. Now consider an arbitrary quadrangle inscribed in a circle. Let be a point on the segment such that . Since the angles and as angles on the same chord are equal, the triangles and are similar, as it was the case for the rectangle. The similarity implies that , i.e. . The triangles and are also similar from the same reasons, and consequently , or that . Taking the two equalities together we get the required result.

Note that [4] ,p. 157-158: “Ptolemy’s theorem has been of eminent importance for the further development of mathematics and its applications. This is because it provides us with the addition theorem for the trigonometric functions. Ptolemy used this theorem for the calculation of tables of chords ... Up to the time of Euler, Ptolemy’s theorem was more important than the law of cosines, the other fundamental generalization of Pythagora’s theorem. After Euler introduced the method of power series, Ptolemy receded into history. With the advent of vector spaces, the law of cosines and its modern equivalent, the scalar product for vectors, came into the foreground. Who can predict the fortunes of contemporary mathematical theories?"

An absolutely amazing reformulation of the original Pythagorean theorem was found by E. W. Dijkstra   :   

Dijkstra’s theorem: If are lengths of the sides of a triangle and are the respective opposite angles then , where denotes the signum function .

Generalizations of Pythagorean theorem to higher dimensions are also known. Here are some of them.

Consider a right-angled tetrahedron , such that in one apex, say , its all three edges meet orthogonally.  If and  are the areas of these three faces, and is the area of the forth one (that is the face opposite to ) then .

A straightforward application of the formula for the area of a triangle and the Pythagorean theorem gives

If is the length of the main diagonal of a rectangular parallelepiped, and are lengths of its sides then .

If is the length of the main diagonal of a rectangular parallelepiped, and the lengths of its face diagonals then .

Notes

References

Cite this web-page as: