Proposition 12
Theorem
If two circles (APB , CPD ) are externally tangent at any point (P), then the line joining the centers must pass through that point.
Commentary
1. Given two circles, let one pass through points A, P, B with center O, and the other pass through points C, P, D with center H.
2. Let these two circles be externally tangent at point P.
3. Then the straight line connecting the two centers O and H passes through the tangent point P that is, the three points O, P, H are collinear.
4. The previous proposition, Book 3 Proposition 11, covers the case when the given two circles are internally tangent.
2. Let these two circles be externally tangent at point P.
3. Then the straight line connecting the two centers O and H passes through the tangent point P that is, the three points O, P, H are collinear.
4. The previous proposition, Book 3 Proposition 11, covers the case when the given two circles are internally tangent.
Original statement
ἐὰν δύο κύκλοι ἐϕάπτωνται ἀλλήλων ἐκτός, ἡ ἐπὶ τὰ κέντρα αὐτῶν ἐπιζϵυγνυμένη διὰ τῆς ἐπαϕῆς ἐλϵύσϵται.
English translation
If two circles touch one another externally, the straight line joining their centres will pass through the point of contact.
