PROPOSITION 34. PROBLEM.
From a given circle to cut off a segment containing an angle equal to a given rectilineal angle.
Let ABC be the given circle, and D the given rectilineal angle: it is required to cut off from the circle ABC a segment containing an angle equal to the angle D.
Draw the straight line BF touching the circle ABC at the point B; [III. 17.
and at the point B,in the straight line BF, make the angle FBC equal to the angle D. [I. 23.
The segment BAC shall contain an angle equal to the angle D.
Because the straight line EF touches the circle ABC, and BC is drawn from the point of contact B, [Constr.
therefore the angle FBC is equal to the angle in the alternate segment BAC of the circle. [III. 32.
But the angle FBC is equal to the angle D. [Construction.
Therefore the angle in the segment BAC is equal to the angle D. [Axiom 1.
Wherefore, from the given circle ABC, the segment BAC has been cut off, containing an angle equal to the given angle D. q.e.f.
PROPOSITION 35. THEOREM.
If two straight lines cut one another within a circle, the rectangle contained hy the segments of one of them shall be equal to the rectangle contained by the segments of the other.
