Proposition 32
Theorem
If a line (EF ) is tangent to a circle, and from the point of tangency (A), a chord (AC ) is drawn, the angles (∠FAC , ∠EAC ) made by this line with the tangent are respectively equal to the angles (∠ABC , ∠ADC ) in the alternate segments of the circle.
Commentary
1. Given a circle O passing through points A, B, C, D, let EF be tangent to the circle at a point A.
2. LetAC be a chord and AB be a diameter of the circle.
3.∠FAC and ∠EAC are made by AC and EF .
4. Then∠FAC = ∠ABC , where ∠ABC is in the segment ABC (the area bounded by arc ABC and AC ), and ∠EAC = ∠ADC , where ∠ADC is in the segment ADC (the area bounded by arc ADC and AC ).
2. Let
3.
4. Then
Original statement
ἐὰν κύκλου ἐϕάπτηταί τις ϵὐθϵῖα, ἀπὸ δὲ τῆς ἁϕῆς ϵἰς τὸν κύκλον διαχθῇ τις ϵὐθϵῖα τέμνουσα τὸν κύκλον, ἃς ποιϵῖ γωνίας πρὸς τῇ ἐϕαπτομένῃ, ἴσαι ἔσονται ταῖς ἐν τοῖς ἐναλλὰξ τοῦ κύκλου τμήμασι γωνίαις.
English translation
If a straight line touch a circle, and from the point of contact there is drawn across, in the circle, a straight line cutting the circle, the angles which it makes with the tangent will be equal to the angles in the alternate segments of the circle.
