PROPOSITION 24.
Similar segments of circles on equal straight lines are equal to one another. For let AEB, CFD be similar segments of circles on equal straight lines AB, CD;I say that the segment AEB is equal to the segment CFD. For, if the segment AEB be applied to CFD, and if the point A be placed on C and the straight line AB on CD,
or it will fall awry, as CGD, and a circle cuts a circle at more points than two : which is impossible. [III. 10] Therefore, if the straight line AB be applied to CD, the segment AEB will not fail to coincide with CFD also;
Therefore etc. Q. E. D. 1