Proposition 35
Theorem
Alternate name(s): intersecting chords theorem.
If two chords (AB , CD ) of a circle intersect at a point (E) within the circle, the rectangles contained by the pieces of the chords are equal (ￜAE ￜ⋅ￜEB ￜ = ￜCE ￜ⋅ￜED ￜ ).
Commentary
1. Let AB and CD be two chords of a given circle and intersect at a point E.
2. ThenￜAE ￜ⋅ￜEB ￜ = ￜCE ￜ⋅ￜED ￜ .
3. Euclid formulated the equation in the previous step geometrically by saying the rectangle with sidesAE and EB (the pieces of chord AB ) is equal to (has the same area as) the rectangle with sides CE and ED (the pieces of chord CD ).
4. This proposition is also known as the intersecting chords theorem and can be summed up as: the products of the lengths of the pieces of two intersecting chords of a circle are equal.
2. Then
3. Euclid formulated the equation in the previous step geometrically by saying the rectangle with sides
4. This proposition is also known as the intersecting chords theorem and can be summed up as: the products of the lengths of the pieces of two intersecting chords of a circle are equal.
Original statement
ἐὰν ἐν κύκλῳ δύο ϵὐθϵῖαι τέμνωσιν ἀλλήλας, τὸ ὑπὸ τῶν τῆς μιᾶς τμημάτων πϵριϵχόμϵνον ὀρθογώνιον ἴσον ἐστὶ τῷ ὑπὸ τῶν τῆς ἑτέρας τμημάτων πϵριϵχομένῳ ὀρθογωνίῳ.
English translation
If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.
