Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/102

78 PROPOSITION 7. THEOREM.

If any point he taken in the diameter of a circle which is not the centre, of all the straight lines which can be drawn from this point to the circumference, the greatest is that in which the centre is, and the other part of the diameter is the least; and, of any others, that which is nearer to the straight line which passes through the centre, is always greater 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 ABCD be a circle and AD its diameter, in which let any point F be taken which is not the centre; let E be the centre: of all the straight lines FB, FC, FG, &c. that can be drawn from F to the circumference, FA, which passes through E, shall be the greatest, and FD, the other part of the diameter AD, shall be the least; and of the others FB shall be greater than FC, and FC than FG.

Join BE, CE, GE. Then, because any two sides of a triangle are greater than the third side, [I. 20. therefore BE, EF are greater than BF.

But BE is equal to AE; [I.Def. 15. therefore AE, EF are greater than BF, that is, AF is greater than BF.

Again, because BE is equal to CE, [I. Definition 15. and EF is common to the two triangles BEF, CEF; the two sides BE, EF are equal to the two sides CE, EF, each to each; but the angle BEF is greater than the angle CEF; therefore the base FB is greater than the base FC. [I. 24. In the same manner it may be shewn that FC is greater than FG.

Again, because GF, FE are greater than EG, [I. 20.