Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/110

86 PROPOSITION 14. THEOREM.

Equal straight lines in a circle are equally distant from the centre: and those which are equally distant from the centre are equal to one another.

Let the straight lines AB, CD in the circle ABDC, be equal to one another: they shall be equally distant from the centre.

Take E, the centre of the circle ABDC; [III. 1. and from E draw EF, EG perpendiculars to AB, CD; [I. 12. and join EA,EC.

Then, because the straight line EF, passing through the centre, cuts the straight line AB, which does not pass through the centre, at right angles, it also bisects it; [III. 3. therefore AF is equal to FB, and AB is double of AF. For the like reason CD is double of CG. But AB is equal to CD; [Hypothesis. therefore AF is equal to CG. [Axiom 7.

And because AE is equal to CE, [I. Definition 15. the square on AE is equal to the square on CE. But the squares on AF, FE are equal to the square on AE, because the angle AFE is a right angle; [I. 47. and for the like reason the squares on CG, GE are equal to the square on CE; therefore the squares on AF, FE are equal to the squares on CG, GE. [Axiom 1. But the square on AF is equal to the square on CG, because AF is equal to CG; therefore the remaining square on FE is equal to the remaining square on GE; [Axiom 3. and therefore the straight line EF is equal to the straight line EG. But straight lines in a circle are said to be equally distant