PROP. VIII. THEOR. If a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, is equal to the square of the straight line which is made up of the whole and that part. Let the straight line AB be divided into any two parts in the point C: four times the rectangle AB, BC, together with the square of AC, shall be equal to the square of the straight line made up of AB and BC together. * * M A * * C BD GK N PRO Produce + AB to D, so that BD be equal + to CB, and upon AD describe + the square AEFD; and construct two figures such as in the preceding. Because CB is equal to BD, and that CB is equal to GK, and BD to KN; therefore GK is equal to KN: for the same reason, PR is equal to RO: and because CB is equal to BD, and GK to KN, the rectangle CK is equal to BN, and GR to RN: but CK is equal to RN, because they are the complements of the parallelogram CO; therefore also BN is equal + to GR: therefore the four rectangles BN, CK, GR, RN, are equal to one another, and so are quadruple of one of them CK. Again, because CB is equal to BD, and that BD is equal to BK, that is †, to CG, and CB equal to GK, that is †, to GP; therefore CG is equal + to GP: and because CG is equal to GP, and PR to RO, the rectangle AG is equal † to MP, and PL to RF: but MP is equal to PL, because they are the complements of the parallelogram ML; wherefore AG is equal + also to RF: therefore the four rectangles AG, MP, PL, RF, are equal to one another, and so are quadruple of one of them AG. And it was demonstrated that the four CK, BN, GR, and RN are quadruple of CK: 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 * E HL F * the rectangle AB, BC, is equal + to the gnomon AOH: +1 Ax. to each of these add XH, which is equal to the square Cor. 4. 2. of AC; therefore four times the rectangle AB, BC, to- and 34. 1. gether with the square of AC, is equal + to the gnomon † 2 Ax. AOH and the square XH: but the gnomon AOH and XH make up the figure AEFD, which is the square of AD: therefore four times the rectangle AB, BC, together with the square of AC, is equal to the square of AD, that is, of AB and BC added together in one straight line. Wherefore, if a straight line, &c. Q. E. D. If a straight line be divided into two equal, and also into two unequal parts; the squares of the two unequal parts are together double of the square of half the line, and of the square of the line between the points of section. Let the straight line AB be divided into two equal parts at the point C, and into two unequal parts at the point D: the squares of AD, DB, shall together be double of the squares of AC, CD. * From the point C draw CE at right angles to AB, *11. 1. and make it equal † to AC or CB, and join EA, EB; † 3.1. through D draw* DF parallel to CE, and through F * 31. 1. draw FG parallel to BA; and join AF. Then, because AC is equal to CE, the angle EAC is equal to the *5. 1. angle AEC: and because the angle ACE is a right angle, the two others AEC, EAC, together make one * *32. 1. right angle; and they are equal to one another; therefore each of them is half of a right angle. For the same reason each of the angles ČEB, 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, and EGF a right angle, for it is equal to the interior and opposite angle ECB, the remaining angle EFG is * * E F CD B * 29. 1. half a right angle: therefore the angle GEF is equal to and FDB a right angle, for it is equal to the inte- * 29. 1. 47.1. † 47.1. * 34. 1. * 47.1. 11. 1. † 3. 1. * 31. 1. *29.1. * 12 Ax. * * And because AC is equal to CE, the square of AC is equal to the square of CE; therefore the squares of AC, CE, are double of the square of AC: but the square of AE is equal to the squares of AC, CE because ACE is a right angle; therefore the square of AE is double of the square of AC: again, because EG is equal to GF, the square of EG is equal to the square of GF; therefore the squares of EG, GF are double of the square of GF: but the square of EF is equal to the squares of EG, GF; therefore the square of EF is double of the square of GF; and GF is equal to CD ; therefore the square of EF is double of the square of CD: but the square of AE is likewise double of the square of AC; therefore the squares of AE, EF are double of the squares of AC, CD: but the square of AF is equal to the squares of AE, EF, because AEF is a right angle; therefore the square of AF is double of the squares of AC, CD: but the squares of AD, DF, are equal to the square of AF, because the angle ADF is a right angle; therefore the squares of AD, DF are double of the squares of AC, CD: and DF is equal to DB; therefore the squares of AD, DB, are double of the squares of AC, CD. If therefore a straight line, &c. Q. E. D. * PROP. X. THEOR. If a straight line be bisected, and produced to any point, the square of the whole line thus produced, and the square of the part of it produced, are together double of the square of half the line bisected, and of the square of 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: the squares of AD, DB shall be double of the squares of AC, CD. From the point C draw* CE at right angles to AB, and make† it equal to AC or CB, and join AE, EB; through E draw* EF parallel to AB, and through D draw DF parallel to CE. And because the straight line EF meets the parallels EC, FD, the angles CEF, EFD are equal * to two right angles; 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 less than two right angles will meet* if produced far enough; therefore EB, FD will * 5.1. meet, if produced, towards B, D: let them meet in G, * * DBG; wherefore also the side BD * * * a * * angle FEG is half a right angle, and therefore equal E 46.1. 10. 1. 3. 1. * 46. 1. * 6.2. * 47.1. † 3 Ax. +30 Def. † 1 Ax. +3Аx. † 30 Def. 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 AB describe the square ABDC; bisect* AC in E, and join BE: produce CA to F, and make* EF equal to EB, and upon AF describe the square FGHA: AB shall be divided in H, so that the rectangle AB, BH is equal to the square of AH. * 1 E HB Produce GH to K: and because the straight line AC is bisected in E, and produced to the point F, the rectangle CF, FA together with the square of AE, is equal to the square of EF: but EF is equal to EB; therefore the rectangle CF, FA, together with the square of AE, is equal to the square of EB: but the squares of BA, AE, are equal to F the square of EB, because the angle EAB is a right angle; therefore the rectangle CF, FA, together with the square of AE, is equal to the squares of BA, AE: take away the square of AE, which is common to both, therefore the remaining rectangle CF, FA, is equal to the square of AB: but 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 HD: but HD is the rectangle contained by AB, BH, for AB is equal † to BD; and FH is the square of AH: therefore the rectangle AB, BH is equal to the square of AH. Wherefore the straight line AB is divided in H, so that the rectangle AB, BH is equal to the square of AH. Which was to be done. PROP. XII. THEOR. K In obtuse angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side pro |