Proposition 8a
Theorem
Alternate name(s): peacock tail.
Commentary
2. Construct a line through P and O.
3. Find two points E and F on the convex circumference and construct line segments
4. Then
5. Book 3 Propositions 8b and 8c deal with the question of when line segments from P to the circumference can be equal in length.
6. Book 3 Proposition 7a is similar except P is inside of the circle.
Original statement
ἐὰν 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 ἐϕ᾽ ἑκάτϵρα τῆς ἐλαχίστης.
English translation
If a point is taken outside a circle and from the point straight lines are 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.
