Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/112

88 but BE, EC are greater than BC; [I. 20.

therefore also AD is greater than BC.

And, because BC is nearer to the centre than FG, [Hypothesis.

EH is less than EK. [III.Def.5.

Now it may be shewn, as in the preceding proposition, that BC is double of BH, and FG double of FK, and that the squares on EH, HB are equal to the squares on EK, KF.

But the square on EH is less than the square on EK, because EH is less than EK;

therefore the square on HB is greater than the square on KF;

and therefore the straight line BH is greater than the straight line FK;

and therefore BC is greater than FG.

Next, let BC be greater than FG: BC shall be nearer to the centre than FG, that is, the same construction being made, EH shall be less than EK.

For, because BC is greater than FG, BH is greater than FK.

But the squares on BH, HE are equal to the squares on FK,KE;

and the square on BH is greater than the square on FK;

because BH is greater than FK;

therefore the square on HE is less than the square on KE;

and therefore the straight line EH is less than the straight line EK.

Wherefore, the diameter &c.

PROPOSITION 16. THEOREM.

The straight line drawn at right angles to the diameter of a circle from the extremity of it, falls without the circle; and no straight line can be drawn from the extremity, between that straight line and the circumference, so as not to cut the circle.