§3.prop.8#1ἐὰν 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.
If 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.
ἔστω κύκλος ὁ ΑΒΓ, καὶ τοῦ ΑΒΓ εἰλήφθω τι σημεῖον ἐκτὸς τὸ Δ, καὶ ἀπʼ αὐτοῦ διήχθωσαν εὐθεῖαί τινες αἱ ΔΑ, ΔΕ, ΔΖ, ΔΓ, ἔστω δὲ ἡ ΔΑ διὰ τοῦ κέντρου.
Let ΑΒΓ be a circle, and of ΑΒΓ let some point Δ be taken outside, and from it let some straight lines ΔΑ, ΔΕ, ΔΖ, ΔΓ be drawn, and let ΔΑ be through the center.
λέγω, ὅτι τῶν μὲν πρὸς τὴν ΑΕΖΓ κοίλην περιφέρειαν προσπιπτουσῶν εὐθειῶν μεγίστη μέν ἐστιν ἡ διὰ τοῦ κέντρου ἡ ΔΑ, μείζων δὲ ἡ μὲν ΔΕ τῆς ΔΖ ἡ δὲ ΔΖ τῆς ΔΓ, τῶν δὲ πρὸς τὴν ΘΛΚΗ κυρτὴν περιφέρειαν προσπιπτουσῶν εὐθειῶν ἐλαχίστη μέν ἐστιν ἡ ΔΗ ἡ μεταξὺ τοῦ σημείου καὶ τῆς διαμέτρου τῆς ΑΗ, ἀεὶ δὲ ἡ ἔγγιον τῆς ΔΗ ἐλαχίστης ἐλάττων ἐστὶ τῆς ἀπώτερον, ἡ μὲν ΔΚ τῆς ΔΛ, ἡ δὲ ΔΛ τῆς ΔΘ.
εἰλήφθω γὰρ τὸ κέντρον τοῦ ΑΒΓ κύκλου καὶ ἔστω τὸ Μ·
I say that of the straight lines falling on the concave circumference ΑΕΖΓ, the greatest is ΔΑ which is through the center, and ΔΕ is greater than ΔΖ, and ΔΖ than ΔΓ; while of the straight lines falling on the convex circumference ΘΛΚΗ, the least is ΔΗ which is between the point and the diameter ΑΗ, and the nearer to the least ΔΗ is always less than the more remote, ΔΚ than ΔΛ, and ΔΛ than ΔΘ.
καὶ ἐπεζεύχθωσαν αἱ ΜΕ, ΜΖ, ΜΓ, ΜΚ, ΜΛ, ΜΘ.
καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΜ τῇ ΕΜ, κοινὴ προσκείσθω ἡ ΜΔ·
For let the center of the circle ΑΒΓ be taken, and let it be Μ; and let ΜΕ, ΜΖ, ΜΓ, ΜΚ, ΜΛ, ΜΘ be joined.
ἡ ἄρα ΑΔ ἴση ἐστὶ ταῖς ΕΜ, ΜΔ. ἀλλʼ αἱ ΕΜ, ΜΔ τῆς ΕΔ μείζονές εἰσιν·
And since ΑΜ is equal to ΕΜ, let ΜΔ be added as common; therefore ΑΔ is equal to ΕΜ, ΜΔ.
καὶ ἡ ΑΔ ἄρα τῆς ΕΔ μείζων ἐστίν.
But ΕΜ, ΜΔ are greater than ΕΔ; therefore ΑΔ is also greater than ΕΔ.
πάλιν, ἐπεὶ ἴση ἐστὶν ἡ ΜΕ τῇ ΜΖ, κοινὴ δὲ ἡ ΜΔ, αἱ ΕΜ, ΜΔ ἄρα ταῖς ΖΜ, ΜΔ ἴσαι εἰσίν· καὶ γωνία ἡ ὑπὸ ΕΜΔ γωνίας τῆς ὑπὸ ΖΜΔ μείζων ἐστίν.
Again, since ΜΕ is equal to ΜΖ, and ΜΔ is common, therefore ΕΜ, ΜΔ are equal to ΖΜ, ΜΔ; and the angle ΕΜΔ is greater than the angle ΖΜΔ.
βάσις ἄρα ἡ ΕΔ βάσεως τῆς ΖΔ μείζων ἐστίν.
Therefore the base ΕΔ is greater than the base ΖΔ.
ὁμοίως δὴ δείξομεν, ὅτι καὶ ἡ ΖΔ τῆς ΓΔ μείζων ἐστίν· μεγίστη μὲν ἄρα ἡ ΔΑ, μείζων δὲ ἡ μὲν ΔΕ τῆς ΔΖ, ἡ δὲ ΔΖ τῆς ΔΓ.
καὶ ἐπεὶ αἱ ΜΚ, ΚΔ τῆς ΜΔ μείζονές εἰσιν, ἴση δὲ ἡ ΜΗ τῇ ΜΚ, λοιπὴ ἄρα ἡ ΚΔ λοιπῆς τῆς ΗΔ μείζων ἐστίν·
Similarly then we shall prove that ΖΔ is also greater than ΓΔ; therefore ΔΑ is the greatest, and ΔΕ is greater than ΔΖ, and ΔΖ than ΔΓ. And since ΜΚ, ΚΔ are greater than ΜΔ, and ΜΗ is equal to ΜΚ, therefore the remainder ΚΔ is greater than the remainder ΗΔ.