fore the rectangle BE, EF, together with the fquare of EG, Book II. is equal to the square of GH: But the fquares of HE, EG

common to both, and the remaining rectangle BE, EF is e-
qual to the fquare of EH: But the rectangle contained by BE,
EF is the parallelogram BD, becaufe EF is equal to ED;
therefore BD is equal to the fquare of EH; but BD is equal
to the rectilineal figure A; therefore the rectilineal figure A
is equal to the fquare of EH: Wherefore a square has been
made equal to the given rectilineal figure A, viz. the fquare
defcribed upon EH.
EH. Which was to be done.

IF

PROP. A. THE O R.

F one fide of a triangle be bifected, the fum of See N. the fquares of the other two fides is double of the fquare of half the fide bifected, and of the square of the line drawn from the point of bisection to the oppofite angle of the triangle.

Let ABC be a triangle, of which the fide BC is bisected in D, and DA drawn to the oppofite angle; the fquares of BA and AC are together double of the fquares of BD and DA.

From A draw AE perpendicular to BC, and because BEA is a right angle, the fquare of AB is equal to the fquares of BE a and EA; for the fame reason, the fquare of AC is equal to the two fquares of CE and EA. Therefore, the fquares of BA and AC are equal to the fquares of BE and EC, together with twice the fquare of EA. But because the

a 47, I.

Book II. line BC is cut equally in D, and unequally in E, the fquares b9.-2. of BE and EC b are equal to twice the fquares of BD and DE; therefore the fquares of BA and AC are equal to twice the fquare of BD, together with twice the fquares of DE and EA. Now, the fquares of DE and EA are équal to the fquare of DA a, and therefore twice the fquares of DE and EA, to twice the fquare: DA. Wherefore alfo, the squares

b 29. 1.

of BA and AC are equal to twice the fquare of BD, together with twice the fquare of DA. Therefore, &c. Q. E. D.

PROP. B. THE OR.

THE
HE fum of the fquares of the diameters of any
parallelogram is equal to the fum of the
fquares of the fides of the parallelogram.

Let ABCD be a parallelogram, of which the diameters are AC and BD; the fum of the fquares of AC and BD is equal to the fum of the fquares of AB, BC, CD, DA.

Let AC and BD interfect one another in E: and because the vertical angles AED, CEB are equal a, and alfo the alternate angles EAD, ECB b, the triangles ADE, CEB have two angles in the one equal to two angles in the other, each to each but the fides AD and BC, which are oppofite to equal angles in these tri

F 34. 1. angles, are also equal c;

therefore the other fides
which are opposite to
the equal angles are

d 26. 1. equald, viz. AE to EC,
and ED to EB.

Since, therefore, BD

is bifected in E, the B

fquares of BA and

AD are equal to

A

D

E

twice the fquare of BE, together with twice the fquare of e A. 2. EA¤; and for the fame reason the square of BC and CD

are

are equal to twice the fquare of BE, together with twice the Book II. fquare of EC, that is, of EA, because EC is equal to EA. Therefore the four fquares of BA, AD, DC, CB are equal to four times the fquares of BE and of EA. But the square of BD is equal to four times the fquare of BE, because BD is double of BEe; and for the fame reason, the fquare of AC e Cor .8.2. is equal to four times the square of AE: Wherefore also, the fquares of BD and AC are equal to the four fquares of BA, AD, DC, CB.

Q. E, D.

Therefore the fum of the fquares, &c.

COR. From the demonftration, it is manifeft that the diameters of every parallelogram bifect one another.