Proposition 19
Theorem
If a line (AB ) is tangent to a circle, then the line (AC ) drawn at right angles to it from the point of tangency (A) passes through the center.
Commentary
1. Given a circle centered at O, let AB be tangent to the circle at point A.
2. LetAC be perpendicular to AB at point A.
3. ThenAC passes through the center O.
4. This proposition is a partial converse of Book 3 Proposition 18.
2. Let
3. Then
4. This proposition is a partial converse of Book 3 Proposition 18.
Original statement
ἐὰν κύκλου ἐϕάπτηταί τις ϵὐθϵῖα, ἀπὸ δὲ τῆς ἁϕῆς τῇ ἐϕαπτομένῃ πρὸς ὀρθὰς γωνίας ϵὐθϵῖα γραμμὴ ἀχθῇ, ἐπὶ τῆς ἀχθϵίσης ἔσται τὸ κέντρον τοῦ κύκλου.
English translation
If a straight line touches a circle, and from the point of contact a straight line is drawn at right angles to the tangent, the centre of the circle will be on the straight line so drawn.
