A swan sits at the centre point of an impeccably round pound. At an edge of the pond stands a monstrous lion holding up to eat up the swan. The lion is afraid of water and, therefore, plan to catch the swan as soon as it reaches the short of the pound. Speed of swan is one-fourth of the speed of lion, moreover the lion always runs in the direction round the shore which brings it closer to the swan the fastest.

Both the swan and the lion can change directions in any given time.

The swan realizes that only chance to escape is to reach the shore without getting caught by the lion and then get into the safe forest lake which is just next to the pound.

By what method can the swan successfully escape?

Let us assume,
radius of pound is R.
speed of beast is S.
speed of lion is S/4.
Circumference = 2 * Pi * R.

Now, if lion swims R/4 distance from center of the pound and then begins to swim across the pond in center implies both lion and beast can take the round trip in same time.
Explanation :
time by beast : 2 * Pi* R * S
time by lion : 2 * Pi* R/4 * S/4 => 2 * Pi* R * S

Now, the lion can move slowly inward toward the center of pound, and begin swimming around the center in a circle from this distance. It is now going around a very slightly smaller circle than it was a moment ago, and thus will be able to swim around this circle FASTER than the beast can run around the shore.

The lion can keep swimming around this way, pulling further away each second, until finally it is on the opposite side of its inner circle from where the beast is on the shore. At this point, the lion aims directly toward the closest shore and begins swimming that way. At this point, the lion has to swim [0.75R feet + 1 millimeter] to get to shore. Meanwhile, the beast will have to run R*pi feet (half the circumference of the pound) to get to where the lion is headed.
The beast runs four times as fast as the lion, but you can see that it has more than four times as far to run:

[0.75R feet + 1 millimeter] * 4 < R*pi

[This math could actually be incorrect if R were very small, but, in that case, we could just say the lion swam inward even less than a millimeter and make the math work out correctly.]

Because the lion has less than a fourth of the distance to travel as the beast, it will reach the shore before the beast reaches where it is and successfully escape.