Hadley's theorem, proposed and proved by Frank Hadley in 1980, is a little-known but pleasing theorem in plane geometry. It is of some academic interest for its resemblance in form to the Pythagorean theorem.

## Definition

A Hadley triangle is an obtuse-angled triangle in which one acute angle is two thirds the complement of the other.

## Example

Typically letting angle C be 2/3 the complement of angle A, one example of a Hadley triangle ABC would have angles A, B and C of 30°, 110° and 40° respectively.

## Theorem

Let ABC be a Hadley triangle in which B the obtuse angle and C is 2/3 the complement of A. Let the respective opposite sides be a, b and c. Then

${\displaystyle b^{2}=c^{2}+ab\,}$

or

${\displaystyle {\frac {b}{c}}={\frac {c}{b-a}}.}$

## Notes

Reminiscent of Pythagoras, in a Hadley triangle, "The square on the longest side is equal to the sum of the square on the first side and the rectangle whose sides are the longest side and the second side."