Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/104

80 convex circumference, the least is that between the point without the circle and the diameter; and of the rest, that which is nearer to the least is always less than one more remote; and from the same point there can he drawn to the circumference two straight lines, and only two, which are equal to one another, one on each side of the shortest line.

Let ABC be a circle, and D any point without it, and from D let the straight lines DA, DE, DF, DC be drawn to the circumference, of which DA passes through the centre: of those which fall on the concave circumference AEFC, the greatest shall be DA which passes through the centre, and the nearer to it shall be greater than the more remote, namely, DE greater than DF, and DF greater than DC; but of those which fall on the convex circumference GKLH, the least shall be DG between the point D and the diameter AG, and the nearer to it shall be less than the more remote, namely, DK less than DL, and DL less than DH.

Take M, the centre of the circle ABC, [III. 1. and join ME, MF, MC, MH, ML, MK.

Then, because any two sides of a triangle are greater than the third side, [I. 20. therefore EM, MD are greater than ED. But EM is equal to AM;[I.Def.15. therefore AM, MD are greater than ED, that is, AD is greater than ED.

Again, because EM is equal to FM, and MD is common to the two triangles EMD, FMD; the two sides EM, MD are equal to the two sides FM, MD, each to each; but the angle EMD is greater than the angle FMD; therefore the base ED is greater than the base FD. [1. 24.

In the same manner it may be shewn that FD is greater than CD.