Problem. About a given square (ABCD) to describe a circle.
Sol.—Draw the diagonals AC, BD intersecting in O (see diagram to Proposition vi.). O is the centre of the required circle.
Dem.—Since ABC is an isosceles triangle, and the angle B is right, each of the other angles is half a right angle; therefore BAO is half a right angle. In like manner ABO is half a right angle; hence the angle BAO equal ABO; therefore [I. vi.] AO is equal to OB. In like manner OB is equal to OC, and OC to OD. Hence the circle described, with O as centre and OA as radius, will pass through the points B, C, D, and be described about the square.