Elements II.11 (idea)

To cut a given straight line so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment.

It is required to cut AB so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment.

Describe the square ABDC on AB. Bisect AC at the point E, and join BE.

Draw CA through to F, and make EF equal to BE.

Describe the square FH on AF, and draw GH through to K.

I say that AB has been cut at H so that the rectangle AB by BH equals the square on AH.

Since the straight line AC has been bisected at E, and FA is added to it, the rectangle CF by FA together with the square on AE equals the square on EF.

But EF equals EB, therefore the rectangle CF by FA together with the square on AE equals the square on EB.

But the sum of the squares on BA and AE equals the square on EB, for the angle at A is right, therefore the rectangle CF by FA together with the square on AE equals the sum of the squares on BA and AE.

Subtract the square on AE from each. Therefore the remaining rectangle CF by FA equals the square on AB.

Now the rectangle CF by FA is FK, for AF equals FG, and the square on AB is AD, therefore FK equals AD.

Subtract AK from each. Therefore FH which remains equals HD.

And HD is the rectangle AB by BH, for AB equals BD, and FH is the square on AH,

therefore the rectangle AB by BH equals the square on HA.

Therefore the given straight line AB has been cut at H so that the rectangle AB by BH equals the square on HA.

Q.E.D.

Proposition 10 <-- Proposition 11 --> Proposition 12