Proposition 18
Theorem
If a line (AB ) is tangent to a circle, then the line (OA ) from the center to the point of tangency is perpendicular to it.
Commentary
1. Given a circle centered at O, let AB be tangent to the circle at point A.
2. ConnectOA .
3. ThenOA is perpendicular to AB at point A.
4. This proposition is a partial converse of Book 3 Proposition 19 and essentially a converse to Book 3 Proposition 16.
2. Connect
3. Then
4. This proposition is a partial converse of Book 3 Proposition 19 and essentially a converse to Book 3 Proposition 16.
Original statement
ἐὰν κύκλου ἐϕάπτηταί τις ϵὐθϵῖα, ἀπὸ δὲ τοῦ κέντρου ἐπὶ τὴν ἁϕὴν ἐπιζϵυχθῇ τις ϵὐθϵῖα, ἡ ἐπιζϵυχθϵῖσα κάθϵτος ἔσται ἐπὶ τὴν ἐϕαπτομένην.
English translation
If a straight line touches a circle, and a straight line is joined from the centre to the point of contact, the straight line so joined will be perpendicular to the tangent.
