Almost Pythagorean

Years ago when I taught in Michigan I often gave talks on a collection of "Almost Pythagorean" relationships... things that reminded me of the classic. At one time I must have had a dozen or so, but I found a new one today on one of the blogs I follow.

Arjen Dijksman is a Dutch Physicist whose blogs often tease out relations between physics and geometry,but his recent post is on the "Lost theorem about angular proportions"

The property applies to any triangle in which one interior angle is twice another. Some beautiful extensions of chords in a unit circle produce a proof, but the part I liked was that if the sides and angles are labeled as shown :

Then a²= c²+bc

addendum, I noticed a couple of days after I posted this, that if you draw a segment call it d, cutting the angle formed by c and a so that c and d form the legs of an isosceles triangle, then the other triangle is also isosceles. One triangle with base angles of 2 theta, and the other with base angles of theta.