The elements of Euclid for the use of schools and colleges : comprising the first two books and portions of the eleventh and twelfth books; with notes and exercises . o the four are quadruple of uae ofthem AG. And it was shewn that the four CK, BN, GR and RN arc quadruple of CK \ therefore the eight rectangleswhich make up the gnomon A OH are quadruple of ^A. And because AK is the rectangle contained by AB, BC,for BK is equal to BC ; therefore four times the rectangle AB, BC is quadrnnloof^^. But the gnomon AOII was she^vn to be quadrupleof^A: Therefore four times the rectangle AB, BC is equal
The elements of Euclid for the use of schools and colleges : comprising the first two books and portions of the eleventh and twelfth books; with notes and exercises . o the four are quadruple of uae ofthem AG. And it was shewn that the four CK, BN, GR and RN arc quadruple of CK \ therefore the eight rectangleswhich make up the gnomon A OH are quadruple of ^A. And because AK is the rectangle contained by AB, BC,for BK is equal to BC ; therefore four times the rectangle AB, BC is quadrnnloof^^. But the gnomon AOII was she^vn to be quadrupleof^A: Therefore four times the rectangle AB, BC is equal to thegnomon AOH. [Axiom 1. Ti) each of these add XH, which is equal to the square onAC. [LI. 4, Corollary, and I. 34. Therefore four times the rectangle AB, BC, together withtlie square on AC, is ecjual to the gnomon AOH and thestjuare XH. J>ut the gnomon AOH and the square XH make up thefigure AEFD, which is the square (^n AD. Therefore four times the rectangle A B. BC, together withthe square on AC, is equal to the square on AD, that is tothe square on the line made of AB and BC together. Wherefore, if a straight line &c. 62 EUCLIDS PROPOSITION 9. THEOREM. If a straight line he divided into two equal, and alsointo two unequal 2^arts, the squares on the two unequalparts are together double of the square on half the lineand of the square on the line between the points of section. Let the straight line AB he divided into two equalparts at the point (7, and into two unequal parts at thepoint D : the squares on AD, DB shall be together doubleof the squares on AC, CD. From the point C drawCE at right angles to AB, [ make it equal to AC orCB, [I. 3. and join EA, EB ; throughD draw DF parallel to CE, andthrough F dmw FG parallelto BA ; [I. 31. and join AF. Then, because AC\^ 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, EACslyq together equ
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