C is the midpoint

As AR, BS, CT are the minimum distances of the line PQ from the points A, B and C, this can be only achieved when AB||PQ and AR, CT and BS are perpendicular to it.

This makes ARTC and CTSB as a parallelogram.

⇒ CT = AR from ARTC parallelogram,

And CT = BS from CTSB parallelogram.

On adding the above two equations, we get

⇒ AR + BS = 2CT