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CD + BC . AD = - CA B D , or AB . CD + 13C . AD + CA . BD = O . Prop. -Ifperpend�'culars be drawnfrom the angular • points of a square to any l�'ne, the sum of the squares of the perpend�'culars from one pa�'r of C oppo8ite ang le8 exceed8 tw�'ce the rectangle of the perpendicularsfrom the other pair of oppo8ite ang le8 by the area of the 8quare. D h Dem. - Let ABCD be the square, L the line ; let fall the 1-8 AM, BN, CP, DQ, on L : through A draw EF 11 to L . Now, L N M P Q since the L BAD is right, the sum ' of the L s BAE, DAF = one right L , and ..

For, let the two first in­ tersect in O. OB = two rightL s ; and BDO + BOC = two right L s ; . ' . the L s AFB, BDC, AOB, COll = four right L s ? and the t. § 38 A SEQUEL TO EUCLID. OB, BOC, COA = four right L s ; . ' . 6 DEF, that is = two right L s ; . . the qua­ drilateral AECO is in­ scribed in a 0 . Hence the three 0 s pass through a common point, which is a given p oint. Again, since the area . of the � DEF is a maxi­ mum, each of its sides is a maximum. Hence ( 1 5) we have to draw through the point A a line 11 to the line joining the centres of the 0 s ABF, CEA ; that is, a line 1- to AO, and join its extremities E, F to the points C, B, respectively.

Which we shall prove, on the properties of the centre of mean position for any system of points and any syst� m of multiples. The Props. ix. and x. of the Second Book are particular cases of this Prop . , viz . , when the point C i s in the line AB or the line AB produce d . -If the base of a 6 be given, both in magnitude and position, and the sum of the squares of the sides in magnitude, the locus of the vertex is a 0 . Prop . -The sum of the squares of the ilz"agonals of a parallelogram is equal to the B sum of the squat'es of its four sides.

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