Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/103

 and that EG is equal to ED; therefore GF, FE are greater than ED. Take away the common part FE, and the remainder GF is greater than the remainder FD. Therefore FA is the greatest, and FD the least of all the straight lines from F to the circumference; and FB is greater than FC, and FC than FG. Also, there can be drawn two equal straight lines from the point F to the circumference, one on each side of the shortest line FD. For, at the point E, in the straight line EF, make the angle FEH equal to the angle FEG, and join FH. Then, because EG is equal to EH, and EF is common to the two triangles GEF, HEF; the two sides EG, EF are equal to the two sides EH, EF, each to each; and the angle GEF is equal to the angle HEF; therefore the base FG is equal to the base FH. But, besides FH, no other straight line can be drawn from F to the circumference, equal to FG. For, if it be possible, let FK be equal to FG. Then, because FK is equal to FG, and FH is also equal to FG, therefore FH is equal to FK; that is, a line nearer to that which passes through the centre is equal to a line which is more remote; which is impossible by what has been already shewn. Wherefore, if any point be taken &c. PROPOSITION 8. THEOREM.

If any point be taken without a circle, and straight lines be drawn from it to the circumference, one of which passes through the centre; of those which fall on the concave circumference, the greatest is that which passes through the centre, and of the rest, that which is nearer to the one passing through the centre is always greater than one more remote; but of those which fall on the