§3.prop.8#2ὥστε ἡ ΗΔ τῆς ΚΔ ἐλάττων ἐστίν·
Therefore ΗΔ is less than ΚΔ.
καὶ ἐπεὶ τριγώνου τοῦ ΜΛΔ ἐπὶ μιᾶς τῶν πλευρῶν τῆς ΜΔ δύο εὐθεῖαι ἐντὸς συνεστάθησαν αἱ ΜΚ, ΚΔ, αἱ ἄρα ΜΚ, ΚΔ τῶν ΜΛ, ΛΔ ἐλάττονές εἰσιν· ἴση δὲ ἡ ΜΚ τῇ ΜΛ· λοιπὴ ἄρα ἡ ΔΚ λοιπῆς τῆς ΔΛ ἐλάττων ἐστίν.
And since, on one of the sides ΜΔ of the triangle ΜΛΔ, two straight lines ΜΚ, ΚΔ have been constructed within, therefore ΜΚ, ΚΔ are less than ΜΛ, ΛΔ; and ΜΚ is equal to ΜΛ; therefore the remainder ΔΚ is less than the remainder ΔΛ.
ὁμοίως δὴ δείξομεν, ὅτι καὶ ἡ ΔΛ τῆς ΔΘ ἐλάττων ἐστίν· ἐλαχίστη μὲν ἄρα ἡ ΔΗ, ἐλάττων δὲ ἡ μὲν ΔΚ τῆς ΔΛ ἡ δὲ ΔΛ τῆς ΔΘ.
λέγω, ὅτι καὶ δύο μόνον ἴσαι ἀπὸ τοῦ Δ σημείου προσπεσοῦνται πρὸς τὸν κύκλον ἐφʼ ἑκάτερα τῆς ΔΗ ἐλαχίστης·
Similarly then we shall prove that ΔΛ is also less than ΔΘ; therefore ΔΗ is the least, and ΔΚ is less than ΔΛ, and ΔΛ than ΔΘ. I say that also only two equal straight lines will fall from the point Δ on the circle on either side of the least ΔΗ.
συνεστάτω πρὸς τῇ ΜΔ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ Μ τῇ ὑπὸ ΚΜΔ γωνίᾳ ἴση γωνία ἡ ὑπὸ ΔΜΒ καὶ ἐπεζεύχθω ἡ ΔΒ. καὶ ἐπεὶ ἴση ἐστὶν ἡ ΜΚ τῇ ΜΒ, κοινὴ δὲ ἡ ΜΔ, δύο δὴ αἱ ΚΜ, ΜΔ δύο ταῖς ΒΜ, ΜΔ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ· καὶ γωνία ἡ ὑπὸ ΚΜΔ γωνίᾳ τῇ ὑπὸ ΒΜΔ ἴση· βάσις ἄρα ἡ ΔΚ βάσει τῇ ΔΒ ἴση ἐστίν.
For let there be constructed with the straight line ΜΔ and at the point Μ on it the angle ΔΜΒ equal to the angle ΚΜΔ, and let ΔΒ be joined. And since ΜΚ is equal to ΜΒ, and ΜΔ is common, therefore the two ΚΜ, ΜΔ are equal to the two ΒΜ, ΜΔ, each to each; and the angle ΚΜΔ is equal to the angle ΒΜΔ; therefore the base ΔΚ is equal to the base ΔΒ.
λέγω, ὅτι τῇ ΔΚ εὐθείᾳ ἄλλη ἴση οὐ προσπεσεῖται πρὸς τὸν κύκλον ἀπὸ τοῦ Δ σημείου.
I say that another straight line equal to the straight line ΔΚ will not fall from the point Δ on the circle.
εἰ γὰρ δυνατόν, προσπιπτέτω καὶ ἔστω ἡ ΔΝ. ἐπεὶ οὖν ἡ ΔΚ τῇ ΔΝ ἐστιν ἴση, ἀλλʼ ἡ ΔΚ τῇ ΔΒ ἐστιν ἴση, καὶ ἡ ΔΒ ἄρα τῇ ΔΝ ἐστιν ἴση, ἡ ἔγγιον τῆς ΔΗ ἐλαχίστης τῇ ἀπώτερον ἴση· ὅπερ ἀδύνατον ἐδείχθη.
For, if possible, let it fall and let it be ΔΝ. Since therefore ΔΚ is equal to ΔΝ, but ΔΚ is equal to ΔΒ, therefore ΔΒ is also equal to ΔΝ, the nearer to the least ΔΗ equal to the more remote; which was proved impossible.
οὐκ ἄρα πλείους ἢ δύο ἴσαι πρὸς τὸν ΑΒΓ κύκλον ἀπὸ τοῦ Δ σημείου ἐφʼ ἑκάτερα τῆς ΔΗ ἐλαχίστης προσπεσοῦνται.
Therefore no more than two equal straight lines will fall from the point Δ on the circle ΑΒΓ on either side of the least ΔΗ.
ἐὰν ἄρα κύκλου ληφθῇ τι σημεῖον ἐκτός, ἀπὸ δὲ τοῦ σημείου πρὸς τὸν κύκλον διαχθῶσιν εὐθεῖαί τινες, ὧν μία μὲν διὰ τοῦ κέντρου αἱ δὲ λοιπαί, ὡς ἔτυχεν, τῶν μὲν πρὸς τὴν κοίλην περιφέρειαν προσπιπτουσῶν εὐθειῶν μεγίστη μέν ἐστιν ἡ διὰ τοῦ κέντρου, τῶν δὲ ἄλλων ἀεὶ ἡ ἔγγιον τῆς διὰ τοῦ κέντρου τῆς ἀπώτερον μείζων ἐστίν, τῶν δὲ πρὸς τὴν κυρτὴν περιφέρειαν προσπιπτουσῶν εὐθειῶν ἐλαχίστη μέν ἐστιν ἡ μεταξὺ τοῦ τε σημείου καὶ τῆς διαμέτρου, τῶν δὲ ἄλλων ἀεὶ ἡ ἔγγιον τῆς ἐλαχίστης τῆς ἀπώτερόν ἐστιν ἐλάττων, δύο δὲ μόνον ἴσαι ἀπὸ τοῦ σημείου προσπεσοῦνται πρὸς τὸν κύκλον ἐφʼ ἑκάτερα τῆς ἐλαχίστης· ὅπερ ἔδει δεῖξαι.
If therefore a point be taken outside a circle, and from the point some straight lines be drawn to the circle, one of which is through the center, and the remaining ones at random, then of the straight lines falling on the concave circumference, the greatest is that through the center, while of the others the nearer to the one through the center is always greater than the more remote; and of the straight lines falling on the convex circumference, the least is that between the point and the diameter, while of the others the nearer to the least is always less than the more remote, and only two equal straight lines will fall from the point on the circle on either side of the least; which was to be proved.