Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/86

62 PROPOSITION 9. THEOREM.

If a straight line he divided into two equals and also into two unequal parts, the squares on the two unequal parts are together double of the square on half the line and of the square on the line between the points of section.

Let the straight line AB be divided into two equal parts at the point C, and into two unequal parts at the point 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, [I. 3. and join EA, EB ; through D draw DF parallel to CE, and through F draw FG parallel to BA ; [I. 31. and join AF.

Then, because AC is equal to CE, [Construction. the angle EAC is equal to the angle AEC. [I. 5. And because the angle ACE is a right angle, [Construction. the two other angles AEC, EAC are together equal to one right angle ; [I. 32. and they are equal to one another ; therefore each of them is half a right angle. For the same reason each of the angles CEB, EBC is half a right angle.

Therefore the whole angle AEB is a right angle.

And because the angle GEF is half a right angle, and the angle EGF a right angle, for it is equal to the interior and opposite angle ECB ; [I. 29. therefore the remaining angle EFG is half a right angle. Therefore the angle GEF is equal to the angle EFG, and the side EG is equal to the side GF. [I. 6. Again, because the angle at B is half a right angle, and the