Reflection

Why is ${\theta}_{i}={\theta}_{r}$?

Refraction

Why is $\frac{{n}_{1}}{{n}_{2}}=\frac{\sin{{\theta}_{1}}}{\sin{\theta}_{2}}$?

Brachistochrone

Why is $(x,y)=(a\theta-a\sin{\theta},a-a\cos{\theta})?$