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 alfo 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 of AB, ac, together with the rectangle of AB, BC, is also equal to the fquare 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 fquare of the aforefaid part.

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 Cв 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 RE is equal to BC (II. Def. 2.)

And, in like manner, AD is a rectangle contained by AC, CP, or by aC, CB; and CE is the square 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 alfo equal to the rectangle of AC, CB together with the fquare 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 thofe 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 square 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 alfo 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 squares 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 fquare of the whole line will be equal to four times the fquare of half the line.

PROP. XII. THEOREM.

If a right line be divided into any two parts, the fquares of the whole line, and one of the parts, are equal to twice the rectangle of the whole line and that part, together with the fquare of the other part.

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

For, upon AB make the fquare AD (II. 1.), and draw the diagonal BE; and make FC, HK parallel to BD, BA(I. 27.):

Then because AG is equal to GD (II. 6.), to each of these equals add CK, and the whole AK will be equal to the whole CD.

And, fince the doubles of equals are equal, the gnomon HBF, together with CK, will be the double of ak.

But CK is a fquare upon CB (II. 7.), and twice the rectangle of AB, BC is the double of AK, whence the gnomon HBF, together with the square CK, is, alfo, equal to twice the rectangle AB, BC.

And,

And, because HF is a fquare upon HG or AC (II. 7.); if this be added to each of these equals, the gnomon HBF, together with the fquares CK, HF, will be equal to twice the rectangle AB, BC, together with the fquare of ac.

But the gnomon HBF, together with the fquares CK, HF, are equal to the whole fquare AD, together with the fquare CK; confequently, the fquares of AB, BC, are equal to twice the rectangle of AB, BC together with the fquare of AC. Q. E. D.

PRO P. XIII. THEOREM.

The difference of the fquares of any two unequal lines, is equal to a rectangle under their fum and difference.

H

Κ

Let AB, AC be any two unequal lines; then will the difference of the fquares of thofe lines be equal to a rectangle under their fum and difference.

For upon AB, AC make the fquares AE, AI (II. 1.); and in HE, produced, take EG equal to AC (I. 3.); and make GF parallel to EB (I. 27.); and produce CI, IK till they meet HG, GF in D and F.

Then, fince HE is equal to AB (Def. II. 2.) and EG to AC (by Conft.), HG will be equal to the fum of AB and AC.