Proposition 9
Theorem
A point (P) within a circle (ABC ) from which three equal lines (PA, PB, and PC ) can be drawn to the circumference is the center.
Commentary
1. Given a circle centered at O, let P be any point inside the circle.
2. From P, suppose three equal linesPA , PB and PC can be drawn to the circumference.
3. Then, P and O must be the same point, meaning P is the center of the circle.
4. This proposition is a logical equivalent of Book 3 Proposition 7c.
2. From P, suppose three equal lines
3. Then, P and O must be the same point, meaning P is the center of the circle.
4. This proposition is a logical equivalent of Book 3 Proposition 7c.
Original statement
ἐὰν κύκλου ληϕθῇ τι σημϵῖον ἐντός, ἀπὸ δὲ τοῦ σημϵίου πρὸς τὸν κύκλον προσπίπτωσι πλϵίους ἢ δύο ἴσαι ϵὐθϵῖαι, τὸ ληϕθὲν σημϵῖον κέντρον ἐστὶ τοῦ κύκλου.
English translation
If a point is taken within a circle, and more than two equal straight lines fall from the point on the circle, the point taken is the centre of the circle.
