Proposition 11
Construction
To cut a given line segment into two pieces so that the rectangle contained by the whole and one of the pieces equals the square of the remaining piece.
Construction Steps
1. Let AB be the given line segment.
2. Construct a square◇ABDC on AB .
3. BisectAC at E and join BE .
4. ExtendCA to F such that ￜEF ￜ = ￜEB ￜ .
5. Construct a square◇AFGH on AF .
6. ExtendGH so it intersects CD at K. The given line segment AB has been cut at H so that the area of the rectangle ◇HBDK equals the area of the square ◇AFGH , ￜAB ￜ⋅ￜBH ￜ = ￜAH ￜ2 .
2. Construct a square
3. Bisect
4. Extend
5. Construct a square
6. Extend
Original statement
τὴν δοθϵῖσαν ϵὐθϵῖαν τϵμϵῖν ὥστϵ τὸ ὑπὸ τῆς ὅλης καὶ τοῦ ἑτέρου τῶν τμημάτων πϵριϵχόμϵνον ὀρθογώνιον ἴσον ϵἶναι τῷ ἀπὸ τοῦ λοιποῦ τμήματος τϵτραγώνῳ.
English translation
To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment.
