Of straight lines in a circle the diameter is greatest, and of the rest the nearer to the center is always greater than the more remote.

Let *ABCD* be a circle, *AD* its diameter, and *E* its center. Let *BC* be nearer to the center *AD,* and *FG* more remote.

I say that *AD* is greatest and *BC* greater than *FG.*

Draw *EH* and *EK* from the center *E* perpendicular to *BC* and *FG.*

Then, since *BC* is nearer to the center and *FG* more remote, *EK* is greater than *EH.*

Make *EL* equal to *EH.* Draw *LM* through *L* at right angles to *EK,* and carry it through to *N.* Join *ME, EN, FE,* and *EG.*

Then, since *EH* equals *EL, BC* also equals *MN.*

Again, since *AE* equals *EM,* and *ED* equals *EN, AD* equals the sum of *ME* and *EN.*

But the sum of *ME* and *EN* is greater than *MN,* and *MN* equals *BC,* therefore *AD* is greater than *BC.*

And, since the two sides *ME* and *EN* equal the two sides *FE* and *EG,* and the angle *MEN* greater than the angle *FEG,* therefore the base *MN* is greater than the base *FG.*

But *MN* was proved equal to *BC.*

Therefore the diameter *AD* is greatest and *BC* greater than *FG.*

Therefore *of straight lines in a circle the diameter is greatest, and of the rest the nearer to the center is always greater than the more remote.*

Q.E.D.