Escaping a circular field guarded by a fast dog

You stand in the middle of a round field whose radius is $R$. A dog that cannot leave the perimeter fence runs four times faster than you. How can you reach the fence at a point where the dog is not waiting?

Circle first, then dash. On a circle of radius just under $R/4$ your angular speed beats the dog's, so you can get to the point diametrically opposite it. From there you need $\tfrac{3R}{4}$ of running, while the dog must cover $\pi R$ at four times your speed, i.e. the equivalent of $\tfrac{\pi R}{4}\approx0.785R>0.75R$, so you arrive first.