Proposition 36
Theorem
If from any point (P) outside a circle, two line segments are drawn to it, one (PT ) of which is a tangent, and the other (PA ) a secant, then the rectangle contained by the secant and the part of the secant outside the circle (PB ) is equal to the square of the tangent (ￜPA ￜ⋅ￜPB ￜ = ￜPT ￜ2 ).
Commentary
1. Given a circle centered at O, let P be a point outside the circle.
2. From P, draw a linePT tangent to the circle at point T.
3. From P, draw a secantPBA which cuts the circle at points B and A.
4. Then the product of the secant (PA ) and the part of the secant outside of the circle (PB ) is equal to the square of the tangent (PT ), that is, ￜPA ￜ⋅ￜPB ￜ = ￜPT ￜ2 .
5. The next proposition, Book 3 Proposition 37, is the converse of this proposition.
2. From P, draw a line
3. From P, draw a secant
4. Then the product of the secant (
5. The next proposition, Book 3 Proposition 37, is the converse of this proposition.
Original statement
ἐὰν κύκλου ληϕθῇ τι σημϵῖον ἐκτός, καὶ ἀπ᾽ αὐτοῦ πρὸς τὸν κύκλον προσπίπτωσι δύο ϵὐθϵῖαι, καὶ ἡ μὲν αὐτῶν τέμνῃ τὸν κύκλον, ἡ δὲ ἐϕάπτηται, ἔσται τὸ ὑπὸ ὅλης τῆς τϵμνούσης καὶ τῆς ἐκτὸς ἀπολαμβανομένης μϵταξὺ τοῦ τϵ σημϵίου καὶ τῆς κυρτῆς πϵριϕϵρϵίας ἴσον τῷ ἀπὸ τῆς ἐϕαπτομένης τϵτραγώνῳ.
English translation
If a point is taken outside a circle and from it there fall on the circle two straight lines, and if one of them cut the circle and the other touch it, the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference will be equal to the square on the tangent.
