Proposition 16
Theorem
A line perpendicular (BI ) to the diameter (AB ) of a circle at one of the diameter's endpoints (B) is tangent to the circle at that point.
Commentary
1. Given a circle centered at O, let AB be the diameter of the circle.
2. LetBI be perpendicular to AB at point B.
3. ThenBI is tangent to the circle at point B.
4. This proposition is essentially a converse of Book 3 Proposition 18.
2. Let
3. Then
4. This proposition is essentially a converse of Book 3 Proposition 18.
Original statement
ἡ τῇ διαμέτρῳ τοῦ κύκλου πρὸς ὀρθὰς ἀπ᾽ ἄκρας ἀγομένη ἐκτὸς πϵσϵῖται τοῦ κύκλου, καὶ ϵἰς τὸν μϵταξὺ τόπον τῆς τϵ ϵὐθϵίας καὶ τῆς πϵριϕϵρϵίας ἑτέρα ϵὐθϵῖα οὐ παρϵμπϵσϵῖται, καὶ ἡ μὲν τοῦ ἡμικυκλίου γωνία ἁπάσης γωνίας ὀξϵίας ϵὐθυγράμμου μϵίζων ἐστίν, ἡ δὲ λοιπὴ ἐλάττων.
English translation
The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed; further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilinear angle.
