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 IG will be equal to the rectangle LC (II. 2.)

But the rectangles HL, LC are, together, equal to the difference of the fquares AE, AI; confequently the rect. angles HL, LG, or the whole rectangle KG, is alfo equal to the difference of those fquares. Q. E. D.

In

PROP. XIV. THEOREM..

any right angled triangle, the fquare of the hypotenufe 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 fquare of the hypotenuse AB be 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 fquares AC, 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.), cc 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, also, 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 fquare AG is double the triangle ABF (I. 32.) and the parallelogram AL is double the triangle ACD (1. 32.); confequently the parallelogram AL is equal to the fquare AG (Ax. 6.)

And, in the same manner, it may be demonstrated, 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.

PRO P. 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 those 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 these 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 square of BD (II. 14.), and the fquares of AC, CB to the fquare of AB (by Hyp.); whence the fquare 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

confequently the angle ACB is equal to the angle BCD (I. 7.)

But the angle BCD is a right angle (by Conft.), whence the angle ACB is also a right angle.

PRO P. XVI. THEOREM.

Q. E. D.

The difference of the fquares of the two fides of any triangle, is equal to the difference of the fquares of the two lines, or distances, included between the extremes of the base and the perpendicular.

A 4

Let ABC be a triangle, having CD perpendicular to AB; then will the difference of the fquares of AC, CB be equal to the difference of the fquares of AD, DB.

For the fum of the fquares of AD, DC is equal to the fquare of AC (II. 14.); and the sum of the fquares of BD, DC is equal to the fquare of BC (II. 14.)

The difference, therefore, between the fum of the fquares of AD, DC, and the sum of the squares of BD, DC is equal to the difference of the fquares of AC, CB.

And, fince DC is common, the difference between the fum of the fquares of AD, DC, and the sum of the squares of BD, DC is equal to the difference of the fquares of AD, DB.

AD, DB.

"

But things which are equal to the fame thing are equal to each other; confequently the difference of the fquares of AC, CB is equal to the difference of the fquares of Q.E.D. COROLL. The rectangle under the fum and difference of the two fides of any triangle, is equal to the rectangle under the base and the difference of the fegments of the base (II. 13.)

PRO P. XVII. THEOREM.

In any obtufe-angled triangle, the square of the fide fubtending the obtufe angle, is greater than the fum of the fquares of the other two fides, by twice the rectangle of the bafe and the distance of the perpendicular from the obtufe angle.

Let ABC be a triangle, of which ABC is an obtufe angle, and CD perpendicular to AB ; then will the fquare of AC be greater than the fquares of AB, BC, by twice the rectangle of AB, BD.

For, fince the right line AD is divided into two parts, in the point B, the fquare of AD is equal to the fquares of