Kitap 3
7
Then, since GE is equal to EH, and EF is common, the two sides GE, EF are equal to the two sides HE, EF; and the angle GEF is equal to the angle HEF; therefore the base FG is equal to the base FH. [I. 4]
7
I say again that another straight line equal to FG will not fall on the circle from the point F.
7
For, if possible, let FK so fall.
7
Then, since FK is equal to FG, and FH to FG, FK is also equal to FH, the nearer to the straight line through the centre being thus equal to the more remote: which is impossible.
7
Therefore another straight line equal to GF will not fall from the point F upon the circle; therefore only one straight line will so fall.
7
Therefore etc. Q. E. D. of the same diameter. I have inserted these words for clearness' sake. The text has simply ἐλαχίστη δὲ ἡ λοιπή, and the remaining (straight line) least. one on each side. The word one is not in the Greek, but is necessary to give the force of ἐφ̓ ἑκάτερα τῆς ἐλαχίστης, literally on both sides, or on each of the two sides, of the least.
PROPOSITION 8.
8
If a point be taken outside a circle and from the point straight lines be drawn through to the circle, one of which is through the centre and the others are drawn at random, then, of the straight lines which fall on the concave circumference, that through the centre is greatest, while of the rest the nearer to that through the centre is always greater than the more remote, but, of the straight lines falling on the convex circumference, that between the point and the diameter is least, while of the rest the nearer to the least is always less than the more remote, and only two equal straight lines will fall on the circle from the point, one on each side of the least.
