A train moving with constant acceleration takes t seconds to pass a certain fixed point and the front and back end of the train pass the fixed point with velocities u and v respectively. Show that the length of the train is $${1 \over 2}(u + v)t$$.

Prove that the centre of the smallest circle passing through origin and whose centre lies on y = x + 1 is $$\left( { - {1 \over 2},{1 \over 2}} \right)$$.

If the area of a rectangle is 64 square units, find the minimum value possible for its perimeter.