PROPOSITION 10. THEOREM.
If a straight line he bisected, and produced to any point, the square on the whole line thus produced, and the square on the part of it produced, are together double of the square on half the line bisected and of the square on the line made up of the half and the part produced.
Let the straight line AB be bisected at C, and pro- duced to D : the squares on AD, DB shall be together double of the squares on AC, CD.
From the point C draw CE at right angles to AB, [I, 11.
and make it equal to AC
or CB; [1.3.
and join AE,EB ; through
E draw EF parallel to
AB, and through D draw
DF parallel to CE. [1.31
Then because the straight
line EF meets the parallels
EC, FD, the angles CEF, CFD are together equal to two
right angles ; [I. 29.
and therefore the angles BEF, EFD are together less
than two right angles.
Therefore the straight lines EB, FD will meet, if produced,
towards B, D, [Axiom 12.
Let them meet at G, and join AG.
Then because AC is equal to CE, [Construction.
the angle CEA is equal to the angle EAC ; [I. 5.
and the angle ACE is a right angle ; [Construction.
therefore each of the angles CEA, EAC is half a right angle. [I. 32.
For the same reason each of the angles CEB, EBC is half
a right angle.
Therefore the angle AEB is a right angle.
And because the angle EBC is half a right angle,
the angle DBG is also half a right angle, for they are verti-
cally opposite ; [I. 15.
but the angle BDG is a right angle, because it is equal to
the alternate angle DCE ; [I. 29.
therefore the remaining angle DGB is half aright angle, [1. 32.
