Kitap 2
9
Now, since AC is equal to CE, the square on AC is also equal to the square on CE; therefore the squares on AC, CE are double of the square on AC.
9
But the square on EA is equal to the squares on AC, CE, for the angle ACE is right; [I. 47] therefore the square on EA is double of the square on AC.
9
Again, since EG is equal to GF, the square on EG is also equal to the square on GF; therefore the squares on EG, GF are double of the square on GF.
9
But the square on EF is equal to the squares on EG, GF; therefore the square on EF is double of the square on GF.
9
But GF is equal to CD; [I. 34] therefore the square on EF is double of the square on CD.
9
But the square on EA is also double of the square on AC; therefore the squares on AE, EF are double of the squares on AC, CD.
9
And the square on AF is equal to the squares on AE, EF, for the angle AEF is right; [I. 47] therefore the square on AF is double of the squares on AC, CD.
9
But the squares on AD, DF are equal to the square on AF, for the angle at D is right; [I. 47] therefore the squares on AD, DF are double of the squares on AC, CD.
9
And DF is equal to DB; therefore the squares on AD, DB are double of the squares on AC, CD.
9
Therefore etc. Q. E. D.
Proposition 10.
10
If a straight line be bisected, and a straight line be added to it in a straight line, the square on the whole with the added straight line and the square on the added straight line both together are double of the square on the half and of the square described on the straight line made up of the half and the added straight line as on one straight line.
10
For let a straight line AB be bisected at C, and let a straight line BD be added to it in a straight line;
10
I say that the squares on AD, DB are double of the squares on AC, CD.
10
For let CE be drawn from the point C at right angles to AB [I. 11], and let it be made equal to either AC or CB [I. 3]; let EA, EB be joined; through E let EF be drawn parallel to AD, and through D let FD be drawn parallel to CE. [I. 31]
