Proposition 11
Theorem
If two circles (APB , CPD ) are internally 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 internally 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 P, O, H are collinear.
4. The next proposition, Book 3 Proposition 12, covers the case when the given two circles are externally tangent.
2. Let these two circles be internally 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 P, O, H are collinear.
4. The next proposition, Book 3 Proposition 12, covers the case when the given two circles are externally tangent.
Original statement
ἐὰν δύο κύκλοι ἐϕάπτωνται ἀλλήλων ἐντός, καὶ ληϕθῇ αὐτῶν τὰ κέντρα, ἡ ἐπὶ τὰ κέντρα αὐτῶν ἐπιζϵυγνυμένη ϵὐθϵῖα καὶ ἐκβαλλομένη ἐπὶ τὴν συναϕὴν πϵσϵῖται τῶν κύκλων.
English translation
If two circles touch one another internally, and their centres are taken, the straight line joining their centres, if it is also produced, will fall on the point of contact of the circles.
