Pythagorean Theorem is one of the most fundamental results of Mathematics. Using the theorem we define what's known as euclidean distance dist₂. This notion of distance extends to spaces with scalar product - Hilbert spaces.

Proofs ##13, 17, and 18 gave us plane generalizations of the theorem. Below I consider another one, now in the 3-dimensional space R³. The statement leads to the definition of euclidean distance in R³. Afterwards, there is an additional and unexpected analog of the theorem in R³.

It's convenient to think of the Pythagorean Theorem as defining the length of the diagonal in a rectangle when its two sides are given. Now consider a parallelepiped with sides a,b, and c. Incidently, the diagonal in question serves as the hypotenuse of the right triangle formed by the edge c and the diagonal of the face ab. The latter, by the Pythagorean Theorem, equals . Applying it the second time gives the length of the diagonal as .

When we moved from a 2-dimensional space to a 3-dimensional space the formula for the diagonal of a shape built on orthogonal segments remained virtually the same except the number of terms grew from 2 to 3, as appropriate. However, in both cases squared were line segments. Pythagorean Theorem has an analog where squared are areas of triangles.

The theorem applies to a special kind of tetrahedra in which all three edges emanating from one of the vertices are perpendicular to each other. One can obtain such a pyramid by cutting a corner from a parallelepiped. Let's introduce areas A,B,C of the faces that house a right angle, and let D be the area of the remaining face. We have

What I want to show is that A² + B² + C² = D².

Draw a plane through p perpendicular to a. Then both k and h will be perpendicular to a. We find that D = ha/2, while h² = k² + p². Also A = ak/2. Therefore,

Q.E.D.

which, in turn, admits a generalization to higher dimensional spaces.

The fact expressed nowadays by the single identity appears in Euclid's Elements as two separate propositions: II.12 for obtuse-angled triangles and II.13 for the acute ones.

Dr. Scott Brodie from the Mount Sinai School of Medicine, NY, sent me a proof of the theorem and a dynamic Geometer's SketchPad illustration.

References

1. G. Polya, Mathematical Discovery, John Wiley and Sons, 1981.

Copyright © 1996-2008 Alexander Bogomolny