Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/83

Rh PROPOSITION 7. THEOREM.

If a straight line he divided into any two parts, the squares on the whole line, and on one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square on the other part.

Let the straight line AB be divided into any two parts at the point C: the squares on AB, BC shall be equal to twice the rectangle AB, BC together with the square on AC.

On AB describe the square ADEB, and construct the figure as in the preceding propositions.

Then AG is equal to GE ; [1. 43. to each of these add CK; therefore the whole AK is equal to the whole CE ; therefore AK, CE are double of AK.

But AK, CE are the gnomon AKF, together with the square CK; therefore the gnomon AKF, together with the square CK, is double of AK. But twice the rectangle AB, BC is double of AK, for BK is equal to BC. [II. 4, Corollary. Therefore the gnomon AKF, together with the square CK, is equal to twice the rectangle AB, BC. To each of these equals add HF, which is equal to the square on AG. [II. 4, Corollary, and I. 34. Therefore the gnomon AKF, together with the squares GK, HF, is equal to twice the rectangle AB, BC, together with the square on AC. But the gnomon AKF together with the squares GK, HF, make up the whole figure ADEB and GK, which are the squares on AB and BC. Therefore the squares on AB, BC, are equal to twice the rectangle AB, BC, together with the square on AC.

Wherefore, if a straight line &c.