Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/120

96 PROPOSITION 24. THEOREM. Similar segments of circles on equal straight lines are equal to one another.

Let AEB, CFD be similar segments of circles on the equal straight lines AB, CD: the segment AEB shall be equal to the segment CFD. For if the segment AEB be applied to the segment CFD, so that the point A may be on the point C, and the straight line AB on the straight line CD, the point B will coincide with the point D, because AB is equal to CD. Therefore, the straight line AB coinciding with the straight line CD, the segment AEB must coincide with the segment CFD; [III. 23. and is therefore equal to it.

Wherefore, similar segments &c.

PROPOSITION 25. PROBLEM.

A segment of a circle being given, to describe the circle of which it is a segment.

Let ABC be the given segment of a circle: it is required to describe the circle of which it is a segment.

Bisect AC at D; [I. 10. from the point D draw DB at right angles to AC; [1. 11. and join AB.

First, let the angles ABD, BAD, be equal to one another. Then DB is equal to DA; [I. 6. but DA is equal to DC; [Construction. therefore DB is equal to DC. [Axiom 1.