Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/116

92 PROPOSITION 19. THEOREM. If a straight line touch a circle, and from the point of contact a straight line he drawn at right angles to the touching line, the centre of the circle shall he in that line.

Let the straight line DE touch the circle ABC at C, and from C let CA be drawn at right angles to DE: the centre of the circle shall be in CA.

For, if not, if possible, let F be the centre, and join CF. Then, because DE touches the circle ABC, and FC is drawn from the centre to the point of contact, FC is perpendicular to DE; [III. 18. therefore the angle FCE is a right angle. But the angle ACE is also a right angle; [Construction.; therefore the angle FCE is equal to the angle ACE, [Ax.11. the less to the greater; which is impossible. Therefore F is not the centre of the circle ABC.

In the same manner it may be shewn that no other point out of CA is the centre; therefore the centre is in CA.

Wherefore, if a straight line &c.

PROPOSITION 20. THEOREM. The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same arc.

Let ABC be a circle, and BEC an angle at the centre, and BAC an angle at the circumference, which have the same arc, BC, for their base: the angle BEC shall be double of the angle BAC.

Join AE, and produce it to F.

First let the centre of the circle be within the angle BAC. Then, because EA is equal to EB, the angle EAB is equal to the angle EBA; [I. 5. therefore the angles EAB, EBA are double of the angle EAB.