In equal circles equal circumferences are subtended by equal straight lines.
IN æqualibus circulis, æquales peripherias, æquales rectæ lineæ subtendunt.
等圜之圜分等。則其割圜分之直線亦等。
Let ABC, DEF be equal circles, and in them let equal circumferences BGC, EHF be cut off;
and let the straight lines BC, EF be joined;
I say that BC is equal to EF.
For let the centres of the circles be taken, and let them be K, L; let BK, KC, EL, LF be joined.
Now, since the circumference BGC is equal to the circumference EHF,
the angle BKC is also equal to the angle ELF. [III. 27]
And, since the circles ABC, DEF are equal, the radii are also equal;
therefore the two sides BK, KC are equal to the two sides EL, LF;
and they contain equal angles;
therefore the base BC is equal to the base EF. [I. 4]
Complete text
Book I