Suppose we have a square ABCD and we draw a circumference which goes through point A and the middle points of the sides BC and CD.
Which has the biggest perimeter, the square or the circle?
An easy way is to look at it as a co-ordinate geometry problem.
Suppose that A is the origin, (with AB as the x-axis), and that the square has a side length of 2.
The point on the circle at the midpoint of DC, call it E, will have co-ordinates (1, 2).
The centre of the circle, call it F, will be on the line AC, suppose that it has co-ordinates (k, k).
Then, if the radius of the circle is r,
AF^2 = FE^2 = r^2,
and using Pythagoras,
k^2 + k^2 = (1 - k)^2 + (2 - k )^2,
2k^2 = 1 - 2k + k^2 + 4 - 4k + k^2,
6k = 5,
k = 5/6.
r^2 = k^2 + k^2,