If a straight line touches a circle, and from the point of contact there is drawn across, in the circle, a straight line cutting the circle, then the angles which it makes with the tangent equal the angles in the alternate segments of the circle.

For let a straight line *EF* touch the circle *ABCD* at the point *B,* and from the point *B* let there be drawn across, in the circle *ABCD,* a straight line *BD* cutting it.

I say that the angles which *BD* makes with the tangent *EF* equal the angles in the alternate segments of the circle, that is, that the angle *FBD* equals the angle constructed in the segment *BAD,* and the angle *EBD* equals the angle constructed in the segment *DCB.*

Draw *BA* from *B* at right angles to *EF,* take a point *C* at random on the circumference *BD,* and join *AD, DC,* and *CB.*

Then, since a straight line *EF* touches the circle *ABCD* at *B,* and *BA* has been drawn from the point of contact at right angles to the tangent, the center of the circle *ABCD* is on *BA.*

Therefore *BA* is a diameter of the circle *ABCD.* Therefore the angle *ADB,* being an angle in a semicircle, is right.

Therefore the sum of the remaining angles *BAD* and *ABD* equals one right angle.

But the angle *ABF* is also right, therefore the angle *ABF* equals the sum of the angles *BAD* and *ABD.*

Subtract the angle *ABD* from each. Therefore the remaining angle *DBF* equals the angle *BAD* in the alternate segment of the circle.

Next, since *ABCD* is a quadrilateral in a circle, the sum of its opposite angles equals two right angles.

But the sum of the angles *DBF* and *DBE* also equals two right angles, therefore the sum of the angles *DBF* and *DBE* equals the sum of the angles *BAD* and *BCD,* of which the angle *BAD* was proved equal to the angle *DBF,* therefore the remaining angle *DBE* equals the angle *DCB* in the alternate segment *DCB* of the circle.

Therefore *if a straight line touches a circle, and from the point of contact there is drawn across, in the circle, a straight line cutting the circle, then the angles which it makes with the tangent equal the angles in the alternate segments of the circle.*

Q.E.D.