Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/128

104. From the demonstration it is manifest that if one angle of a triangle bo equal to the other two, it is a right angle. For the angle adjacent to it is equal to the same two angles; [I. 32. and when the adjacent angles are equal, they are right angles. [I. Definition 10.

PROPOSITION 32. THEOREM.

If a straight line touch a circle, and froam the point of contact a straight line he drawn cutting the circle, the angles which this line makes with the line touching the circle shall be equal to the angles which are in the alternate segments of the circle.

Let the straight line EF touch the circle ABCD at the point B, and from the point B let the straight line BD be drawn, cutting the circle: the angles which BD makes with the touching line EF, shall be equal to the angles in the alternate segments of the circle; that is, the angle DBF shall be equal to the angle in the segment BAD, and the angle DBE shall be equal to the angle in the segment BCD

From the point B draw BA at right angles to EF, [1. 11. and take any point C in the, arc BD, and join AD, DC, CB.

Then, because the straight line EF touches the circle ABCD at the point B, [Hyp. and BA is drawn at right angles to the touching line from the point of contact B, [Construction. therefore the centre of the circle is in BA. [III. 19. Therefore the angle ADB, being in a semicircle, is a right angle. [III. 31. Therefore the other two angles BAD, ABD are equal to a right angle. [I. 32. But ABF is also a right angle. [Construction.