Proposition 37

Theorem

If from any point (P) outside a circle, two line segments are drawn to it, where one (PT ) meets the circle, the other (PA ) is a secant, and the rectangle contained by the secant and the part of the secant outside the circle (PB ) is equal to the square of the line segment that meets the circle (ￜPA ￜ⋅ￜPB ￜ = ￜPT ￜ²), then the line segment which meets the circle is a tangent.

Commentary

1. Given a circle centered at O, let P be a point outside the circle.
2. From P, draw a line PT intersecting the circle at a point T.
3. From P, draw a secant PBA which cuts the circle at points B and A.
4. Suppose the product of the secant (PA ) and the part of the secant outside of the circle (PB ) is equal to the square of the line segment (PT ) that intersects the circle (PT ), that is, ￜPA ￜ⋅ￜPB ￜ = ￜPT ￜ².
5. Then PT is tangent to the circle.
6. The previous proposition, Book 3 Proposition 36, is the converse of this proposition.

Original statement

ἐὰν κύκλου ληϕθῇ τι σημϵῖον ἐκτός, ἀπὸ δὲ τοῦ σημϵίου πρὸς τὸν κύκλον προσπίπτωσι δύο ϵὐθϵῖαι, καὶ ἡ μὲν αὐτῶν τέμνῃ τὸν κύκλον, ἡ δὲ προσπίπτῃ, ᾖ δὲ τὸ ὑπὸ τῆς ὅλης τῆς τϵμνούσης καὶ τῆς ἐκτὸς ἀπολαμβανομένης μϵταξὺ τοῦ τϵ σημϵίου καὶ τῆς κυρτῆς πϵριϕϵρϵίας ἴσον τῷ ἀπὸ τῆς προσπιπτούσης, ἡ προσπίπτουσα ἐϕάψϵται τοῦ κύκλου.

English translation

If a point is taken outside a circle and from the point there fall on the circle two straight lines, if one of them cut the circle, and the other fall on it, and if further 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 is equal to the square on the straight line which falls on the circle, the straight line which falls on it will touch the circle.

Under Development

Last updated on: 2022-09-12.

Proposition 37

Theorem

Commentary

Original statement

English translation

Computable version

Additional instances

Dependency graphs