User:Extcetc/Hadley's Theorem

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


 * $$ b^2 = c^2 + ab \, $$

or


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