Book 3
33
Now, since AD is drawn at right angles to the diameter AE from its extremity, AD touches the circle AEB. [III. 16, Por.]
33
And AB has been drawn across from the point of contact at A; therefore the angle BAD is equal to the angle constructed in the alternate segment AHB of the circle. [III. 32]
33
But the angle BAD is equal to the angle at C.
33
Therefore the angle in the segment AHB is also equal to the angle at C.
33
Therefore on the given straight line AB the segment AHB of a circle has been described admitting an angle equal to the angle at C. Q. E. F.
PROPOSITION 34.
34
From a given circle to cut off a segment admitting an angle equal to a given rectilineal angle.
34
Let ABC be the given circle, and the angle at D the given rectilineal angle; thus it is required to cut off from the circle ABC a segment admitting an angle equal to the given rectilineal angle, the angle at D.
34
Let EF be drawn touching ABC at the point B, and on the straight line FB, and at the point B on it, let the angle FBC be constructed equal to the angle at D. [I. 23]
34
Then, since a straight line EF touches the circle ABC, and BC has been drawn across from the point of contact at B, the angle FBC is equal to the angle constructed in the alternate segment BAC. [III. 32]
34
But the angle FBC is equal to the angle at D; therefore the angle in the segment BAC is equal to the angle at D.
34
Therefore from the given circle ABC the segment BAC. has been cut off admitting an angle equal to the given rectilineal angle, the angle at D. Q. E. F.
PROPOSITION 35.
35
If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.
35
For in the circle ABCD let the two straight lines AC, BD cut one another at the point E; I say that the rectangle contained by AE, EC is equal to the rectangle contained by DE, EB.
