Proposition 5
Theorem
If two circles (ABC , ABD ) cut one another at any point (A), then they are not concentric.
Commentary
1. Given two circles centered at O and P, let these two circles intersect at a point A.
2. Then the two circles are not concentric, meaning O and P are two distinct points.
3. The next proposition, Book 3 Proposition 6, handles the alternative case when two circles "touch" or are tangent to one another.
4. Book 3 Proposition 10 covers the case when two circles have more than two points in common.
2. Then the two circles are not concentric, meaning O and P are two distinct points.
3. The next proposition, Book 3 Proposition 6, handles the alternative case when two circles "touch" or are tangent to one another.
4. Book 3 Proposition 10 covers the case when two circles have more than two points in common.
Original statement
ἐὰν δύο κύκλοι τέμνωσιν ἀλλήλους, οὐκ ἔσται αὐτῶν τὸ αὐτὸ κέντρον.
English translation
If two circles cut one another, they will not have the same centre.
