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 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 square upon CB (II. 7.), and twice the rectangle AB, BC is the double of AK, whence the gnomon HBF, together with the fquare 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 thefe equals, the gnomon HBF, together with the fquares CK, HF, will be equal to twice the rectangle AB, BC, together with the square of ac. But the gnomon HBF, together with the fquares CK, HF, are equal to the whole square AD, together with the fquare CK; confequently, the fquares of AB, BC, are equal to twice the rectangle AB, BC together with the fquare of AC. Q. E. D. PROP. XIII. THEOREM. The difference of the fquares of any two unequal lines, is equal to a rectangle under their fum and difference. 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 squares AE, A1 (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. And because AH is equal to AB, and AK to AC (II. Def. 2.), KH will be equal to CB, or the difference of AB and AC. But the rectangle KG is contained by HG, and HK, whence it is, alfo, contained by the fum and difference of AB and AC. And, fince LE is equal to HK (I. 30.) or CB (by Conft.), and EG to AC (by Conft.) CI, or LB, the rectangle LG will be equal to the rectangle LC (II. 2.) But the rectangles HL, LC are, together, equal to the difference of the squares AE, AI; consequently the rectangles HL, LG, or the whole rectangle KG, is also equal to the difference of those squares. Q. E. D. PROP. XIV. THEOREM. In any right angled triangle, the fquare of the hypotenuse is equal to the fum of the fquares of the other two fides. Let ABC be a right angled triangle, having the right angle ACB; then will the square of the hypotenuse AB he equal to the fum of the fquares of AC and CB. For, on AB, defcribe the fquare AE (II. 1.), and on AC, CB the squares AG, BH; and, through the point c, draw draw CL parallel to AD or BE (I. 27.); and join BF, CD, AK and CE. Then, fince the right line AC meets the two right lines GC, CB in the point c, and makes each of the angles ACG, ACB a right angle (by Hyp. and Def. 2.), GC will be in the fame right line with CB (I. 14.) And, because the angle FAC is equal to the angle DAB (I. 8.), if the angle CAB be added to each of them, the whole angle FAB will be equal to the whole angle DAC. The fides FA, AB, are, alfo, equal to the fides CA, AD, each to each, (Def. 2.), and their included angles have, likewise, been shewn to be equal; whence the triangle ABF is equal to the triangle ACD (I. 4.) But the square AG is double the triangle ABF (I. 32.) · and the parallelogram AL is double the triangle ACD (I 32.); confequently the parallelogram AL is equal to the fquare AG (Ax. 6.) And, in the same manner, it may be demonftrated, that the parallelogram BL is equal to the fquare BH; therefore the whole fquare AE is equal to the fquares AG and BH taken together. Q. E. D. COROLL. The difference of the fquares of the hypotenuse and either of the other fides is equal to the square of the remaining fide. PROP. XV. THEOREM. If the fquare of one of the fides of a triangle be equal to the fum of the fquares of the other two fides, the angle contained by thofe fides will be a right angle. Let ABC be a triangle; then if the fquare of the fide AB be equal to the fum of the fquares of AC, CB, the angle ACB will be a right angle. For, at the point c, make CD at right angles to CB (I. 11.), and equal to AC (I. 3.); and join DB. Then, fince the fquares of equal lines are equal (II. 2.), the fquare of DC will be equal to the square of AC. And, if, to each of thefe equals, there be added the fquare of CB, the fquares of DC, CB will be equal to the fquares of AC, CB. But the fquares of DC, CB are equal to the fquare of BD (II. 14.), and the fquares of AC, CB to the fquare of AB (by Hyp.); whence the square of BD is equal to the fquare of AB. And fince equal fquares have equal fides (II. 3.), AB is equal to BD; BC is alfo common to each of the triangles ABC, DBC, and AC is equal to CD (by Conft.); con |