Proposition 6
Theorem
If one circle (ABC ) is internally tangent to another circle (ADE ) at any point (A), then the two circles are not concentric.
Commentary
1. Let two circles centered at O and P respectively, be tangent to or "touch" each other at a point A.
2. Then the two circles are not concentric, meaning O and P are two distinct points.
3. Euclid only proved the case for two circles being internally tangent. The externally tangent case is trivial.
4. The previous proposition, Book 3 Proposition 5, handles the alternative case when two circles "cut" one another.
5. 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. Euclid only proved the case for two circles being internally tangent. The externally tangent case is trivial.
4. The previous proposition, Book 3 Proposition 5, handles the alternative case when two circles "cut" one another.
5. Book 3 Proposition 10 covers the case when two circles have more than two points in common.
Original statement
ἐὰν δύο κύκλοι ἐϕάπτωνται ἀλλήλων, οὐκ ἔσται αὐτῶν τὸ αὐτὸ κέντρον.
English translation
If two circles touch one another, they will not have the same centre.
