If a point is taken outside a circle and from the point straight lines are drawn through to the circle, one of which is through the center and the others are drawn at random, then, of the straight lines which fall on the concave circumference, that through the center is greatest, while of the rest the nearer to that through the center is always greater than the more remote, but, of the straight lines falling on the convex circumference, that between the point and the diameter is least, while of the rest the nearer to the least is always less than the more remote; and only two equal straight lines fall on the circle from the point, one on each side of the least.

Let *ABC* be a circle, and let a point *D* be taken outside *ABC.* Let straight lines *DA, DE, DF,* and *DC* be drawn through from *D,* and let *DA* be drawn through the center.

I say that, of the straight lines falling on the concave circumference *AEFC,* the straight line *DA* through the center is greatest, while *DE* is greater than *DF,* and *DF* greater than *DC.* But, of the straight lines falling on the convex circumference *HLKG,* the straight line *DG* between the point and the diameter *AG* is least, and the nearer to the least *DG* is always less than the more remote, namely *DK* is less than *DL,* and *DL* is less than *DH.*

Take the center *M* of the circle *ABC.* Join *ME, MF, MC, MK, ML,* and *MH.*

Then, since *AM* equals *EM,* add *MD* to each, therefore *AD* equals the sum of *EM* and *MD.*

But the sum of *EM* and *MD* is greater than *ED,* therefore *AD* is also greater than *ED.*

Again, since *ME* equals *MF,* and *MD* is common, therefore *EM* and *MD* equal *FM* and *MD,* and the angle *EMD* is greater than the angle *FMD,* therefore the base *ED* is greater than the base *FD.*

Similarly we can prove that *FD* is greater than *CD.* Therefore *DA* is greatest, while *DE* is greater than *DF,* and *DF* is greater than *DC.*

Next, since the sum of *MK* and *KD* is greater than *MD,* and *MG* equals *MK,* therefore the remainder *KD* is greater than the remainder *GD,* so that *GD* is less than *KD.*

And, since on *MD,* one of the sides of the triangle *MLD,* two straight lines *MK* and *KD* are constructed meeting within the triangle, therefore the sum of *MK* and *KD* is less than the sum of *ML* and *LD.* And *MK* equals *ML,* therefore the remainder *DK* is less than the remainder *DL.*

Similarly we can prove that *DL* is also less than *DH.* Therefore *DG* is least, while *DK* is less than *DL,* and *DL* is less than *DH.*

I say also that only two equal straight lines will fall from the point *D* on the circle, one on each side of the least *DG.*

Construct the angle *DMB* equal to the angle *KMD* on the straight line *MD* and at the point *M* on it. Join *DB.*

Then, since *MK* equals *MB,* and *MD* is common, the two sides *KM* and *MD* equal the two sides *BM* and *MD* respectively, and the angle *KMD* equals the angle *BMD,* therefore the base *DK* equals the base *DB.*

I say that no other straight line equal to the straight line *DK* falls on the circle from the point *D.*

Above

For, if possible, let a straight line so fall, and let it be *DN.* Then, since *DK* equals *DN,* and *DK* equals *DB, DB* also equals *DN,* that is, the nearer to the least *DG* equal to the more remote, which was proved impossible.

Therefore no more than two equal straight lines fall on the circle *ABC* from the point *D,* one on each side of *DG* the least.

Therefore *if a point is taken outside a circle and from the point straight lines are drawn through to the circle, one of which is through the center and the others are drawn at random, then, of the straight lines which fall on the concave circumference, that through the center is greatest, while of the rest the nearer to that through the center is always greater than the more remote, but, of the straight lines falling on the convex circumference, that between the point and the diameter is least, while of the rest the nearer to the least is always less than the more remote; and only two equal straight lines fall on the circle from the point, one on each side of the least.*

Q.E.D.

This proposition is not used in the rest of the *Elements.*