Kitap 2
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]
10
Then, since a straight line EF falls on the parallel straight lines EC, FD, the angles CEF, EFD are equal to two right angles; [I. 29] therefore the angles FEB, EFD are less than two right angles.
10
But straight lines produced from angles less than two right angles meet; [I. Post. 5] therefore EB, FD, if produced in the direction B, D, will meet.
10
Let them be produced and meet at G, and let AG be joined.
10
Then, since AC is equal to CE, the angle EAC is also equal to the angle AEC; [I. 5] and the angle at C is right; therefore each of the angles EAC, AEC is half a right angle. [I. 32]
10
For the same reason each of the angles CEB, EBC is also half a right angle; therefore the angle AEB is right.
10
And, since the angle EBC is half a right angle, the angle DBG is also half a right angle. [I. 15] But the angle BDG is also right, for it is equal to the angle DCE, they being alternate; [I. 29] therefore the remaining angle DGB is half a right angle; [I. 32] therefore the angle DGB is equal to the angle DBG, so that the side BD is also equal to the side GD. [I. 6]
