Similar segments of circles on equal straight lines equal one another.

Let *AEB* and *CFD* be similar segments of circles on equal straight lines *AB* and *CD.*

I say that the segment *AEB* equals the segment *CFD.*

For, if the segment *AEB* is superposed on *CFD,* and if the point *A* is placed on *C* and the straight line *AB* on *CD,* then the point *B* coincides with the point *D,* because *AB* equals *CD,* and, *AB* coinciding with *CD,* the segment *AEB* also coincides with *CFD.*

For, if the straight line *AB* coincides with *CD* but the segment *AEB* does not coincide with *CFD,* then it either falls within it, or outside it, or it cuts across it, as *CGD,* and a circle cuts a circle at more points than two, which is impossible.

Therefore, if the straight line *AB* is superposed on *CD,* then the segment *AEB* does not fail to coincide with *CFD* also, therefore it coincides with it and equals it.

Therefore *similar segments of circles on equal straight lines equal one another. *

Q.E.D.

This proposition is used in III.26.