A4 same reason each of the angles CEB, EBC is half a right angle. Therefore the angle AEB is a right angle. And because the angle EBC is half a right angle, the angle DBC is also half a right angle, for they are vertically opposite; [I. 15.] but the angle BDG is a right angle, because it is equal to the alternate angle DCE; [I. 29.] therefore the remaining angle DGB is half a right angle, [I. 32.] and is therefore equal to the angle DBG; therefore also the side DB is equal to the side DG. [I. 6.] Again, because the angle EGF is half a right angle, and the angle at Fa right angle, for it is equal to the opposite angle ECD; [I. 34.] therefore the remaining angle FEG is half a right angle, [I. 32.] and is therefore equal to the angle EGF: therefore also the side GF is equal to the side FE. [I. 6.] And because EC is equal to CA, the square on EC is equal to the square on CA; therefore the squares on EC, CA are double of the square on CA. But the square on AE is equal to the squares on EC, CA. [I. 47.] Therefore the square on AE is double of the square on AC. Again, because GF is equal to FE, the square on GF is equal to the square on FE; therefore the squares on GF, FE are double of the square on FE. But the square on EG is equal to the squares on GF, FE. [I. 47.] Therefore the square on EG is double of the square on FE. And FE is equal to CD; [I. 34.] therefore the square on EG is double of the square on CD. But it has been demonstrated that the square on AE is double of the square on AC. Therefore the squares on AE, EG are double of the squares on AC, CD. But the square on AG is equal to the squares on AE, EG. [I. 47.] Therefore the square on AG is double of the squares on AC, CD. But the squares on AD, DG are equal to the square on AG. [I. 47.] Therefore the squares on AD, DG are double of the squares on AC, CD. And DG is equal to DB; therefore the squares on AD, DB are double of the squares on AC, CD. Therefore, if a straight line, &c. Q.E.D. This proposition is identical with proposition 9, and both may be included in one enunciation. By consulting the note to proposition 5 of this book it will be seen that the resulting proposition is, The square on the sum of two lines together with the square on their difference is equal to twice the sum of the squares on the two lines. Ex. If AC contains m lineal units, and CD n such units, shew that this proposition may be thus stated, (m + n)2 + (m − n)2=2m2 + 2n2. and prove that the equation is true. PROPOSITION 11. PROBLEM. To divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts may be equal to the square on the other part. Let AB be the given straight line: it is required 1. to divide it into two parts, so that the rectangle contained by the whole and one of the parts may be equal to the square on the other part. On AB describe the AFGH. AB shall be divided F 3. Then, because the straight line AC is bisected at E, and produced to F, the rectangle CF, FA, together with the square on AE, is equal to the square on EF. [II. 6.] But EF is equal to EB. [Const.] Therefore the rectangle CF, FA, together with the square on AE, is equal to the square on EB. But the square on EB is equal to the squares on AE, AB, because the angle EAB is a right angle. [I. 47.] Therefore the rectangle CF, FA, together with the square on AE, is equal to the squares on AE, AB. Take away the square on AE, which is common to both, therefore the remainder, the rectangle CF, FA, is equal to the square on AB. [Ax. 3.] But the figure FK is the rectangle_contained by CF, FA, for FG is equal to FA; and AD is the square on AB; therefore FK is equal to AD. Take away the common part AK, and the remainder FH is equal to the remainder HD. [Ax. 3.] But HD is the rectangle contained by AB, BH, for AB is equal to BD; and FH is the square on AH; therefore the rectangle AB, BH is equal to the square on AH. Therefore the straight line AB is divided at H, so that the rectangle AB, BH is equal to the square on AH. Q.E.F. A line so divided is sometimes said to be cut in 'medial section.' PROPOSITION 12. THEOREM. In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle, by twice the rectangle contained by the side on which, when produced, the perpendicular falls, and the staight line intercepted without the triangle, between the perpendicular and the obtuse angle. Let ABC be an obtuse-angled triangle, having 1, 2. the obtuse angle ACB, and from the point A let AD be drawn perpendicular to BC produced: the square on AB shall be greater than the squares on AC, CB, by twice the rectangle BC, CD. Because the straight 3. line BD is divided into two parts at the point C, the square on BD is equal to the squares on BC, CD, and twice the rectangle BC, CD. [II. 4.] To each of these equals add the square on DA. Therefore the squares on BD, DA are equal to the squares on BC, CD, DA, and twice the rectangle BC, CD. [Ax. 2.] But the square on BA is equal to the squares on BD, DA, because the angle at D is a right angle; [I. 47.] and the square on CA is equal to the squares on CD, DA. [I 47.] Therefore the square on BA is equal to the squares on BC, CA, and twice the rectangle BC, CD; that is, the square on BA is greater than the squares on BC, CA by twice the rectangle BC, CD. Therefore, in obtuse-angled triangles, &c. Q.E.D. A Ex. 1. If BCA is two-thirds of two right angles, and the triangle is isosceles, shew that the square on BA will be four times the square on BC. 2. If the angle BAC=135°, and BC contains 5 units and CA 8 units, shew that BA will contain very nearly 12 units. PROPOSITION 13. THEOREM. In every triangle, the square on the side subtending an acute angle, is less than the squares on 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 on it from the opposite angle, and the acute angle. Let ABC be any triangle, and the angle at B 1,2. an acute angle; and on BC one of the sides containing it, let fall the perpendicular AD from the opposite angle upon BC or BC produced: the square on AC, opposite to the angle B, shall be less than the squares on CB, BA, by twice the rectangle CB, BD. Then in each case the squares on CB, BD 3. are equal to twice the rectangle contained by CB, BD and the square on CD. [II. 7.] To each of these equals add the square on DA. Therefore the squares on CB, BD, DA are equal to twice the rectangle CB, BD and the squares on CD, DA. [Ax. 2.] But the square on AB is equal to the squares on BD, DA, because the angle BDA is a right angle; [I. 47.] and the square on AC is equal to the squares on CD, DA. [I. 47.] Therefore the squares on CB, BA are equal to the square on AC and twice |