A and C are the opposite vertices of a square ABCD, and have
coordinates (a,b) and (c,d), respectively. What are the coordinates
of the vertices B and D? What is the area of the square?

Fermat's Poser

Age 14 to 16 Challenge Level

Fermat was an amateur mathematician of great intellect. He wrote
this problem:

Find the point whose sum of distances from the vertices
(corners) of a given triangle is a minimum.