To circumscribe a circle about a given square.

Let *ABCD* be the given square.

It is required to circumscribe a circle about the square *ABCD.*

Join *AC* and *BD,* and let them cut one another at *E.*

Then, since *DA* equals *AB,* and *AC* is common, therefore the two sides *DA* and *AC* equal the two sides *BA* and *AC,* and the base *DC* equals the base *BC,* therefore the angle *DAC* equals the angle *BAC.*

Therefore the angle *DAB* is bisected by *AC.*

Similarly we can prove that each of the angles *ABC, BCD,* and *CDA* is bisected by the straight lines *AC* and *DB.*

Now, since the angle *DAB* equals the angle *ABC,* and the angle *EAB* is half of the angle *DAB,* and the angle *EBA* half of the angle *ABC,* therefore the angle *EAB* also equals the angle *EBA,* so that the side *EA* also equals *EB.*

Similarly we can prove that each of the straight lines *EA* and *EB* equals each of the straight lines *EC* and *ED.*

Therefore the four straight lines *EA, EB, EC,* and *ED* equal one another.

Therefore the circle described with center *E* and radius one of the straight lines *EA,* *EB, EC,* or *ED* also passes through the remaining points, and it is circumscribed about the square *ABCD.*

Let it be circumscribed, as *ABCD.*

Therefore a circle has been circumscribed about the given square.

Q.E.F.