Proposition 2
Theorem
If any two points (A, B) are taken on the circumference of a circle, then the segment of the infinite line that lies between these points (A, B) falls inside the circle and the remaining parts of the line are outside the circle.
Commentary
1. Given a circle centered at C, let A and B be two random points on the circumference of the circle.
2. ConnectAB and extend it both ways, so that infinite line EABD is constructed.
3. Then, the line segmentAB falls inside the circle, while the half lines BD and AE fall outside the circle.
4. The second conclusion was not included in Euclid's original statement. It is a corollary of the first conclusion.
2. Connect
3. Then, the line segment
4. The second conclusion was not included in Euclid's original statement. It is a corollary of the first conclusion.
Original statement
ἐὰν κύκλου ἐπὶ τῆς πϵριϕϵρϵίας ληϕθῇ δύο τυχόντα σημϵῖα, ἡ ἐπὶ τὰ σημϵῖα ἐπιζϵυγνυμένη ϵὐθϵῖα ἐντὸς πϵσϵῖται τοῦ κύκλου.
English translation
If on the circumference of a circle two points are taken at random, the straight line joining the points will fall within the circle.
