x 92 PR E IL F therefore the eight rectangles which contain the gnomon AOH, are quadruple of AK. And because AK is the rectangle contained by AB, BC, for BK is equal to BC; therefore four times the rectangle AB, BC is quadruple of AK: but the gnomon AOH was demonstrated to be quadruple of AK; therefore four times the rectangle AB, BC' is equal to the gnomon AOH; to each of these equals add XH, which is equal to the square on AC; therefore four times the rectangle AB, BC, together with the square on AC, is equal to the gnomon AOH and the square XH; but the gnomon AOH and XH make up the figure AEFD, which is EUCLID'S ELEMENTS. the square on AD; therefore four times the rectangle AB, BC together with the square on AC, is equal to the square on AD, that is, on AB and BC added together in one straight line. Wherefore, if a straight line, &c, Q.E.D. PROPOSITION IX. THEOREM. If a straight line be divided into two equal, and also into two unequal parts; the squares on the two unequal parts are together double of the square on half the line, and of the square on the line between the points of section. Let the straight line AB be divided into two equal parts in the point C, and into two unequal parts in the point D. Then the squares on AD, DB together, shall be double of the squares on AC, CD. A C D B From the point C draw CE at right angles to AB, (1. 11.) make CÊ equal to AC or CB, (1. 3.) and join EA, EB; through D draw DF parallel to CE, meeting EB in F, (1. 31.) through F draw FG parallel to BA, and join AF. Then, because AC is equal to CE, therefore the angle AEC is equal to the angle EAC; (1. 5.) therefore the two other angles AEC, EAC of the triangle are together equal to a right angle; (1. 32.) and since they are equal to one another; therefore each of them is half a right angle. For the same reason, each of the angles CEB, EBC is half a right angle; and therefore the whole AEB is a right angle. And because the angle GEF is half a right angle, for it is equal to the interior and opposite angle ECB, (1. 29.) Again, because the angle at B is half a right angle, for it is equal to the interior and opposite angle ECB, (1. 29.) the square on AC is equal to the square on CE; therefore the squares on AC, CE are double of the square on AC; but the square on AE is equal to the squares on AC, CE, (1. 47.) because ACE is a right angle; 5 therefore the square on AE is double of the square on AC. the square on EG is equal to the square on GF; therefore the squares on AE, EF are double of the squares on AC, CD; but the square on AF is equal to the squares on AE, EF, therefore the square on AF'is double of the squares on AC, CD: but the squares on AD, DF are equal to the square on AF; because the angle ADF is a right angle; (1. 47.) therefore the squares on AD, DF are double of the squares on 4 C, CD; and DF is equal to DB; therefore the squares on AD, DB are double of the squares on AC, CD. If therefore a straight line be divided, &c. Q.E. D. PROPOSITION X. THEOREM. If a straight line be bisected, and produced to any point, the square on the whole line thus produced, and the square on the part of it produced, are together double of the square on half the line bisected, and of the square on the line made up of the half and the part produced. Let the straight line AB be bisected in C, and produced to the point D. Then the squares on AD, DB, shall be double of the squares on AC, CD. E F A B D G From the point C draw CE at right angles to AB, (1. 11.) and through D draw DF parallel to CE, meeting EF in F. a Then because the straight line EF meets the parallels CE, FD, therefore the angles CEF, EFD are equal to two right angles; (1. 29.) and therefore the angles BEF, EFD are less than two right angles. But straight lines, which with another straight line make the interior angles upon the same side of a line, less than two right angles, will meet if produced far enough; (I. ax. 12.) therefore EB, FD will meet, if produced towards B, D; therefore the angle CEA is equal to the angle EAC; (1. 5.) each of the angles CEB, EBC is half a right angle; D because it is equal to the alternate angle DCE; (1. 29.) wherefore also the side BD is equal to the side DG. (1. 6.) (I. wherefore also the side GF is equal to the side FE. 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 EA is equal to the squares on EC, CA; (1. 47.) therefore the square on EA 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; (1. 47.) therefore the square on EG is double of the square on FE; and FE is equal to CD; (1. 34.) wherefore the square on EG is double of the square on CD; but it was demonstrated, that the square on EA is double of the square on AC; therefore the squares on EA, EG are double of the squares on AC, CD; but the square on AG is equal to the squares on ÉA, EG; (1. 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; therefore the squares on AD, DG are double of the squares on AC, CD; but DG is equal to DB; therefore the squares on AD, DB are double of the squares on AC, CD. Wherefore, if a straight line, &c. Q. E. D. PROPOSITION XI. PROBLEM. 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 on the other part. Let AB be the given straight line. It is required to divide AB into two parts, so that the rectangle contained by the whole line and one of the parts, shall be equal to the square on the other part. F A E G H B C K D Upon AB describe the square ACDB; (1. 46.) produce CA to F, and make EF equal to EB, (1. 3.) Then AB shall be divided in H, so that the rectangle AB, BH is equal to the square on AH. Produce GH to meet CD in K. Then because the straight line AC'is bisected in E, and produced to F, therefore 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; therefore the rectangle CF, FA together with the square on AE, is equal to the square on EB; but the squares on BA, AE are equal to the square on EB, (1. 47.) because the angle EAB is a right angle; therefore the rectangle CF, FA, together with the square on AE, is equal to the squares on BA, AE; take away the square on AE, which is common to both; therefore the rectangle contained by CF, FA is equal to the square on BA. But the figure FK is the rectangle contained by CF, FA, and AD is the square on AB; therefore the figure FK is equal to AD; take away the common part AK, therefore the remainder FH is equal to the remainder HD; and FH is the square on AH; therefore the rectangle AB, BH, is equal to the square on AH. Wherefore the straight line AB is divided in H, so that the rectangle AB, BH is equal to the square on AH. Q. E. F. PROPOSITION XII. 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 upon which, when produced, the perpendicular falls, and the straight line intercepted 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 perpendicular to BC produced. Then the square on AB shall be greater than the squares on AC, CB, by twice the rectangle BC, CD. A B с D Because the straight line BD is divided into two parts in the point C, therefore 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; but the square on BA is equal to the squares on BD, DA, (1. 47.) because the angle at D is a right angle; and the square on CA is equal to the squares on CD, DA; 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. PROPOSITION XIII. THEOREM. In every triangle, the square on the side subtending either of the acute angles, 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 acute angle and the perpendicular let fall upon it from the opposite 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 AD from the opposite angle. (1. 12.) Then the square on AC opposite to the angle B, shall be less than the squares on CB, BA, by twice the rectangle CB, BD. A |