on Mar 12, 2018
The Pythagorean Theorem is both very useful and widely known. However, it is simply a special case of a much broader theorem, that very few people know.
Consider this isosceles triangle. If a perpendicular bisector from the apex has length a, then we have the familiar Pythagorean Theorem:
a2 + b2 = c2 |  |
But what if the line from the apex is not a perpendicular bisector? If it splits the opposite side into segments of lengths b and d, we have a much more general result, which is due to Larry Hoehn:
a2 + bd = c2 |  |
In the special case where the line segment is a perpendicular bisector, b=d, and we have the Pythagorean case.
Hoehn’s theorem does not mean Pythagoras was wrong (there are more than 400 known proofs of the Pythagorean Theorem), but math teachers should consider teaching Hoehn’s Theorem in conjunction with its Pythagorean sibling.
For more information, see www.cut-the-knot.org/pythagoras/Hoehn.shtml.