TITLE: A geometry problem - easy with trigonometry, harder without it
QUESTION [5 upvotes]: Consider a $\triangle ABC$, with $\angle A=15^{\circ}, \angle B=55^{\circ}, \angle C=110^{\circ}$.
Prove that $c^2=ab+b^2$.
This is from my math teacher. I solved it in 10 minutes with trigonomery, not that difficult. My question is: it is possible to prove it without trigonometry? Please help! I am thankful for every solution!

REPLY [5 votes]: All that matters is that $\angle C = 2 \angle B$:


$$\triangle ABC \sim \triangle AB^\prime B \qquad\to\qquad \frac{c}{b}=\frac{a+b}{c}$$