In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; further the angle of the greater segment is greater than a right angle, and the angle of the less segment is less than a right angle.

Let *ABCD* be a circle, let *BC* be its diameter, and *E* its center. Join *BA, AC, AD,* and *DC.*

I say that the angle *BAC* in the semicircle *BAC* is right, the angle *ABC* in the segment *ABC* greater than the semicircle is less than a right angle, and the angle *ADC* in the segment *ADC* less than the semicircle is greater than a right angle.

Join *AE,* and carry *BA* through to *F.*

Then, since *BE* equals *EA,* the angle *ABE* also equals the angle *BAE.* Again, since *CE* equals *EA,* the angle *ACE* also equals the angle *CAE.* Therefore the whole angle *BAC* equals the sum of the two angles *ABC* and *ACB.*

But the angle *FAC* exterior to the triangle *ABC* also equals the sum of the two angles *ABC* and *ACB.* Therefore the angle *BAC* also equals the angle *FAC.* Therefore each is right. Therefore the angle *BAC* in the semicircle *BAC* is right.

Next, since in the triangle *ABC* the sum of the two angles *ABC* and *BAC* is less than two right angles, and the angle *BAC* is a right angle, the angle *ABC* is less than a right angle. And it is the angle in the segment *ABC* greater than the semicircle.

Next, since *ABCD* is a quadrilateral in a circle, and the sum of the opposite angles of quadrilaterals in circles equals two right angles, while the angle *ABC* is less than a right angle, therefore the remaining angle *ADC* is greater than a right angle. And it is the angle in the segment *ADC* less than the semicircle.

I say further that the angle of the greater segment, namely that contained by the circumference *ABC* and the straight line *AC,* is greater than a right angle, and the angle of the less segment, namely that contained by the circumference *ADC* and the straight line *AC,* is less than a right angle.

This is at once manifest. For, since the angle contained by the straight lines *BA* and *AC* is right, the angle contained by the circumference *ABC* and the straight line *AC* is greater than a right angle.

Again, since the angle contained by the straight lines *AC* and *AF* is right, the angle contained by the straight line *CA* and the circumference *ADC* is less than a right angle.

Therefore *in a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; further the angle of the greater segment is greater than a right angle, and the angle of the less segment is less than a right angle.*

Q.E.D.

The part of this proposition which says that an angle inscribed in a semicircle is a right angle is often called *Thale’s theorem*.

This proposition is used in III.32 and in each of the rest of the geometry books, namely, Books IV, VI, XI, XII, XIII. It is also used in Book X.