the square of Ac: Again, because GF is equal to FE, the square of GF is equal to the square of FE; and therefore the squares G, FE are double of the square of EF: But the square of EG is equal (1. 47.) to the squares of GF, FE; therefore the square of EG is double of the square of EF: And EF is equal to CD; wherefore the square of Eg is double of the square of cD: But it was demonstrated, that the square of EA is double of the square of AC; therefore the squares of A E, E G are double of the squares of AC, CD: And the square of AG is equal (1. 47.) to the squares of A E, EG: therefore the square of AG is double of the squares of AC, CD: But the squares of AD, DG are equal (1. 47.)to the square of AG; therefore the squares of AD, DG are double of the squares of AC, CD: But dg is equal to DB; therefore the squares of AD, DB are double of the squares of AC, CD: Wherefore, if a straight line, &c. Q. E. D. PROP. XI. PROB. To divide a given straight line into two parts, so that the rectangle contained by the whole, and one of the parts, shall be equal to the square of the other part. Let AB be the given straight line; it is required to divide it into two parts, so that the rectangle contained by the whole, and one of the parts, shall be equal to the square of the other part. Upon A B describe (1. 46.) the square ABDC; bisect (1. 10.) AC in E, and join BE; produce ca to F, and make (1. 3.) EF equal to EB, and upon AF describe (I. 46.) the square FGHA; AB is divided in H, so that the rectangle A B, Bu is equal to the square of AH. Produce go to K: Because the straight line Ac is F G A B bisected in E, and produced to the point F, the rect. the of AE, which is common to both, therefore the remaining rectangle CF, FA is equal to the square of AB: and the figure Fk is the rectangle contained by CF, FA, for AF is equal to FG ; and Ad is the square of AB; therefore Fk is equal to AD: Take away the common part Ak, and the remainder Fh is equal to the remainder ud: And hd is the rectangle contained by AB, BH, for AB is equal to BD; and Fh is the square of AH. Therefore the rectangle A B, Bh is equal to the square of Ah: Wherefore the straight line AB is divided in H, so that the rectangle A B, Bh is equal to the square of AH. Which was to be done? K PROP. XII. THEOR. In obtuse angled triangles, if a perpendicular be drawn from any of the acute angles to the opposite side produced, the square of the side subtending the obtuse angle is greater than the squares of the sides containing the obtuse angle, by twice the rectangle contained by the side upon usħich, when produced, the perpendicular falls, and the straight line inter А В C D cepted without the triangle between the perpendicular and the obtuse angle. Let ABC be an obtuse angled triangle, having the obtuse angle ACB, and from the point a let AD be drawn (1. 12.), perpendicular to B C 'produced: the square of AB is greater than the squares of AC, CB by twice the rectangle BC, C D. Because the straight line en is divided into two parts in the point c, the square of BD is equal (11. 4.) to the squares of BC, CD, and twice the rectangle BC, CD: To each of these equals add the square of DA; and the squares of DB, DA are equal to the squares of BC, CD, DA, and twice the rectangle BC, CD: But the square of BA is equal (1. 47.) to the squares of BD, DA, because the angle at D is a right angle; and the square of cA is equal (I. 47.) to the squares of CD, DA: Therefore the square of BA is equal to the squares of BC, CA, and twice the rectangle BC, CD; that is, the square of B A is greater than the squares of BC, CA, by twice the rectangle BC, CD. Therefore, in obtuse angled triangles, &c. Q. E. D. PROB. XIII. THEOR. In every triangle, the square of the side subtending any of the acute angles is less than the squares of the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall upon it from the opposite angle, and the acute angle. Let ABC be any triangle, and the angle at B one of its acute angles, and upon BC, one of the sides containing it, let fall the perpendicular (1. 12.) AD A B D from the opposite angle : The square of Ac opposite to the angle B, is less than the squares of CB, BA by twice the rectangle CB, BD. First, let ad fall within the triangle ABC; and because the straight line CB is divided into two parts in the point D, the squares of CB, BD are equal (11. 7.) to twice the rectangle contained by CB, BD, and the square of DC: To each of these equals add the square of ad; therefore the squares of CB, BD, DA are equal to twice the rectangle CB, BD, and the squares of AD, DC: But the square of AB is equal (1. 47.) to the squares of BD, DA, because the angle bda is a right angle ; and the square of Ac is equal to the squares of AD, DC: Therefore the squares of CB, BA are equal to the square of AC, and twice the rectangle CB, BD, that is, the square of ac alone is less than the squares of CB, BA by twice the rectangle CB, BD. Secondly, let ad fall without the triangle ABC: Then, because the angle at d is a right angle, the angle ACB is greater (1. 16.) than a right angle: and therefore the square of AB is equal (11. 12.) to the squares of AC, CB, and twice the rectangle BC, CD: To these equals add the square of BC, and the squares of A B, BC are equal to the square of Ac, and twice the square of BC, and twice the rectangle, BC, CD: But because it is divided into two parts in c, the rectangle DB, Bc is equal (11. 3.) to the rectangle BC, CD and the square of Bc: And the doubles of these are equal: Therefore the squares of AB, BC are equal to the square of BC, and twice the rectangle DB, BC: Therefore the square of AC alone A B C D A B is less than the squares of A B, BC by twice the rectangle D B, BC. Lastly, let the side ac be perpendicular to BC; then is bc the straight line between the perpendicular and the acute angle at B; and it is manifest that the squares of A B, BC are equal (1. 47.) to the square of A C, and twice the square of BC: Therefore in every triangle, &c. Q.E. D. PROP. XIV. PROB. To describe a square that shall be equal to a given rectilineal figure. Let A be the given rectilineal figure; it is required to describe a square that shall be equal to A. Describe (1. 45.) the rectangular parallelogram BCDE equal to the rectilineal figure A. If then the sides of it EL BE, ED are one E D equal to another, it is a square, and B what was required is now done: But if they are not equal, produce one of them BE to to F, and make EF equal to ED, and bisect BF in G; and from the centre G, at the distance GB, or GF, describe the semicircle, BHF, and produce DE to H, and join Gh. Then, because the straight line BF is divided into two equal parts in the point G, and into two unequal at E, the rectangle BE, EF, together with the square of EG, is equal (11. 5.) to the square of But GF is equal to Gh; therefore the rectangle EE, EF, together with the square of EG, is GF: |