Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/79

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On BC describe the square CDEB; [I. 46.

produce ED to F, and through A draw AF parallel to CD or BE. [I. 31.

Then the rectangle AE is equal to the rectangles AD, CE.

But AE is the rectangle contained by AB, BC, for it is contained by AB, BE, of which BE is equal to BC;

and AD is contained by AC, CB, for CD is equal to CB;

and CE is the square on BC.

Therefore the rectangle AB, BC is equal to the rectangle AC, CB, together with the square on BC.

Wherefore, if a straight line &c.

PROPOSITION 4. THEOREM. If a straight line be divided into any two parts, the square on the whole line is equal to the squares on the two parts, together with twice the rectangle contained by the two parts.

Let the straight line AB be divided into any two parts at the point C: the square on AB shall be equal to the squares on AC, CB, together with twice the rectangle contained by AC, CB.

On AB describe the square ADEB; [I. 46.

join BD; through C draw CGF parallel to AD or BE, and through G draw HK parallel to AB or DE. [I. 31.

Then, because CF is parallel to AD, and BD falls on them, the exterior angle CGB is equal to the interior and opposite angle ADB; [I. 29.

but the angle ADB is equal to the angle ABD, [I. 5.

because BA is equal to AD, being sides of a square; therefore the angle CGB is equal to the angle CBG; [Ax. 1.

and therefore the side CG is equal to the side CB. [I. 6.

But CB is also equal to GK, and CG to BK; [I. 34.

therefore the figure CGKB is equilateral.