But the fide AH is equal to the fide EK, and the fide HK to the fide AE (I. 30.); whence the figure HE is equilateral.

It has also all its angles right angles:

For EAH is a right angle, being the angle of a square; and HG, EF are each of them parallel to the fides of the fame fquare, whence the remaining angles will also be right angles (I. 25.)

The figure HE, therefore, being equilateral, and having all its angles right angles, is a fquare: and the same may be proved of the figure FG. Q. E. D.

PROP. VIII. THEOREM.

The rectangles contained under a given line and the several parts of another line, any how divided, are, together, equal to the rectangle of the two whole lines,

Let A and BC be two right lines, one of which, BC, is divided into several parts in the points D, E; then will the rectangle of A and BC, be equal to the fum of the rectangles of A and BD, A and DE, and A and EC.

For make BF perpendicular to BC (I. 11.) and equal to A (I. 3.), and draw FG parallel to BC, and DH, EI

and CG each parallel to BF (I. 27.), producing them till they meet FG in the points H, I, G.

Then, fince the rectangle BH is contained by BD and BF (II. Def. 3.), it is alfo contained by BD and A, becaufe BF is equal to A (by Conft.)

And, fince the rectangle DI is contained by DE and DH, it is also contained by DE and A, because DH is equal to BF (I. 30.), or a.

The rectangle EG, in like manner, is contained by EC and A; and the rectangle BG by BC and A.

But the whole rectangle BC, is equal to the rectangles BH, DI and EG, taken together; whence the rectangle of A and BC is also equal to the rectangles of A and BD, A and DE and A and EC, taken together.

PRO P. IX. THEOREM.

Q. E. D.

If a right line be divided into any two parts, the rectangles of the whole line and each of the parts, are, together, equal to the fquare of the whole line.

Let the right line AB be divided into any two parts in the point c; then will the rectangle of AB, AC, together with the rectangle of AB, BC, be equal to the fquare of AB.

For, upon AB defcribe the fquare AD (II. 1.), and through c draw CF parallel to AE or BD (I. 27.)

Then, fince the rectangle AF is contained by AE, AC, it is also contained by AB, AC, because AE is equal to AB (II. Def. 2.)

And, fince the rectangle CD is contained by BD, bc, it is also contained by AB, BC, because BD is equal to AB. But AD, or the fquare of AB, is equal to the rectangles AF, CD, taken together; whence the rectangle AB, AC, together with the rectangle AB, BC, is alfo equal to the square of AB. Q. E. D.

PROP. X. THEOREM.

If a right line be divided into any two parts, the rectangle of the whole line and one of the parts, is equal to the rectangle of the two parts, together with the square of the aforefaid part.

D

Let the right line AB be divided into any two parts in the point c; then will the rectangle of AB, BC be equal to the rectangle of AC, CB, together with the fquare of CB.

For upon CB defcribe the fquare CE (II. 1.), and through a draw AF parallel to CD (I. 27.), meeting ED, produced, in F.

Then, fince AE is a rectangle, contained by AB, BE, it is also contained by AB, BC, because BE is equal to BC (II. Def. 2.)

And, in like manner, AD is a rectangle contained by AC, CD, or by AC, CB; and CE is the fquare of CB (by Conft.)

But the rectangle AE is equal to the rectangle AD, and the fquare CE, taken together; whence the rectangle of AB, BC is also equal to the rectangle of AC, CB together with the square of CB. Q. E. D.

PROP. XI. THEOREM,

If a right line be divided into any two parts, the fquare of the whole line will be equal to the fquares of the two parts, together with twice the rectangle of those parts.

Let the right line AB be divided into any two parts in the point c; then will the square of AB be equal to the fquares of AC, CB together with twice the rectangle of AC, CB.

For upon AB make the fquare AD (II. 1.), and draw the diagonal EB; and make CK, FH parallel to AE, ed (I, 27.) :

Then,

Then, fince the parallelograms about the diagonal of a fquare are themselves fquares (II. 7.), FK will be the fquare of FG, or its equal AC, and CH of CB,

And fince the complements of the parallelograms about the diagonal are equal to each other (II. 6.), the complement AG will be equal to the complement GD.

But AG is equal to the rectangle of AC, CB, because CG is equal to CB (II. Def. 2.); and GD is also equal to the rectangle of AC, CB, because GK is equal to GF (Def. II. 2.) or AC (I. 30.), and GH to CB (I. 30.)

The two rectangles AG, GD are, therefore, equal to twice the rectangle of AC, CB; and FK, CH have been proved to be equal to the fquares of AC, CB.

But these two rectangles, together with the two squares, make up the whole fquare AD; confequently the fquare AD is equal to the fquares of AC, CB, together with twice the rectangle of AC, CB. Q. E. D. COROLL. If a line be divided into two equal parts, the square of the whole line will be equal to four times the fquare of half the line.