Iquare of CE, together with the rectangle of AE, EB be equal to the fquare of AC or CB. For bifect the base AB in D (I. 10.), and join the points E, D. Then, fince AC is equal to CB, AD to DB, and CD is common to each of the triangles ACD, BCD, the angle CDA will be equal to the angle CDB (I. 7.); and confequently CD will be perpendicular to AB (Def. 8, 9.) And, because ACE is a triangle, and CD is the perpendicular, the difference of the fquares of AC, CE is equal to the difference of the fquares of AD, DE (II. 16.) But, fince BE is the fum of AD and DE, and AE is their difference, the difference of the fquares of AD, DE is equal to the rectangle of AE, EB; confequently, the difference of the fquares of AC, CE is alfo equal to the rectangle AE, EB. And if, to each of these equals, there be added the fquare of CE, the fquare of AC will be equal to the íquare of CE, together with the rectangle of AE, EB. PROP. XXI. THEOREM. The diagonals of any parallelogram bifect each other, and the fum of their fquares is equal to the fum of the fquares of the four fides of the parallelogram. Let ABCD be a parallelogram, whofe diagonals AC, BD interfect each other in E; then will AE be equal to EC, and BE to ED; and the fum of the fquares of AC, -BD will be equal to the fum of the fquares of AB, BC, CD and DA. For fince AB, DC are parallel, and AC, BD interfect them, the angle DCE will be equal to the angle EAB (I. 24.), and the angle CDE to the angle EBA (I. 24:) The angle DEC is likewife equal to the angle AEB (I. 15.), and the fide DC to the fide AB (I. 30.); confequently DE is also equal to EB, and CE to EA (I. 21.) Again, fince DB is bifected in E, the sum of the squareş of DC, CB will be equal to twice the fum of the fquares of DE, EC (II. 19.) And, because DC is equal to AB, and CB to'DA (I. 30.) the fum of the fquares of AB, CB, DC and DA are equal to four times the fum of the fquares of DE, EC. But four times the fquare of DE is equal to the fquare of BD (II. 11. Cor.), and four times the fquare of Ee is equal to the square of AC; whence the fum of the squares of AC, BD are equal to the fum of the fquares of AB, BC, CD and DA. Q. E. D. PROP. XXII. PROBLEM To divide a given right line into two parts, fo that the rectangle contained by the whole line and one of the parts, shall be equal to the fquare of the other part. Let AB be the given right line; it is required to divide it into two parts, fo that the rectangle of the whole line and one of the parts fhall be equal to the fquare of the other part. Upon AB defcribe the fquare AC (II. 1.), and bifect the fide of it AD in E (I. 10.) Join the points B, E; and, in EA produced, take EF equal to EB (I. 3.); and upon AF defcribe the fquare FH (II. 1.) Then will AB be divided in H fo, that the rectangle AB, BH, will be equal to the fquare of AH. For, fince DF is equal to the fum of EB and ED, or its equal EA, and AF is equal to their difference, the rectangle of DF, FA is equal to the difference of the fquares of EB, EA (II. 13.) But the rectangle of DF, FA is equal to DG, Decause FA is equal to FG (II. Def. 2.); and the difference of the fquares of EB, EA is equal to the fquare of AB (II. 14. Cor.); whence DG is equal to AC. And, if from each of these equals, the part DH, which is common to both, be taken away, the remainder AG will be equal to the remainder HC. But HC is the rectangle of AB, BH; for AB is equal to BC; and AG is the fquare of AH; therefore the right line AB is divided in H fo, that the rectangle of AB, Bн is equal to the fquare of AH, which was to be done, BOOK III. DEFINITION S. 1. A radius of a circle, is a right line drawn from the centre to the circumference. 2. A diameter of a circle, is a right line drawn through the centre and terminated both ways by the circumference. 3. An arc of a circle, is any part of its periphery, or circumference. о 4. The chord, or fubtenfe, of an arc, is a right line joining the two extremities of that arc. 5. A femicircle, is a figure contained under any diameter and the part of the circumference cut off by that diameter. e |