Proposition 7c
Theorem
Commentary
2. Connect
3. Let E and F be two points on the circumference with
4. Find a point G such that
5. Then G cannot lie on the circumference, which means at most two equal line segments can be drawn from P to the circumference.
6. Book 3 Proposition 8c is similar except P is outside of the circle.
7. This proposition is a logical equivalent of Book 3 Proposition 9.
Original statement
ἐὰν κύκλου ἐπὶ τῆς διαμέτρου ληϕθῇ τι σημϵῖον, ὃ μή ἐστι κέντρον τοῦ κύκλου, ἀπὸ δὲ τοῦ σημϵίου πρὸς τὸν κύκλον προσπίπτωσιν ϵὐθϵῖαί τινϵς, μϵγίστη μὲν ἔσται, ἐϕ᾽ ἧς τὸ κέντρον, ἐλαχίστη δὲ ἡ λοιπή, τῶν δὲ ἄλλων ἀϵὶ ἡ ἔγγιον τῆς διὰ τοῦ κέντρου τῆς ἀπώτϵρον μϵίζων ἐστίν, δύο δὲ μόνον ἴσαι ἀπὸ τοῦ σημϵίου προσπϵσοῦνται πρὸς τὸν κύκλον ἐϕ᾽ ἑκάτϵρα τῆς ἐλαχίστης.
English translation
If on the diameter of a circle a point is taken which is not the centre of the circle, and from the point straight lines fall upon the circle, that will be greatest on which the centre is, the remainder of the same diameter will be least, and of the rest the nearer to the straight line through the centre is always greater than the more remote, and only two equal straight lines will fall from the point on the circle, one on each side of the least straight line.
