H draw KLM parallel to AD or EF, and also through A draw AK parallel to CL or DM: and be A C B D cause AC is equal to CB, the rectangle AL is equal (43. 1.) to CH; L M but CH is equal (36. 1.) to HF; therefore also AL is equal to HF: to each of these add CM; therefore the whole AM is equal to the E G F gnomon CMG: and DM is the rectangle contained by AD, DB, for DM is equal (Cor. 4. 2.) to DB : therefore the gnomon CMG is equal to the rectangle AD, DB, add to each of these LG, which is equal to the square of CB, therefore the rectangle AD, DB, together with the square of CB, is equal to the gnomon CMG and the figure LG: but the gnomon CMG and LG make up the whole figure CEFD, which is the square of CD; therefore the rectangle AD, DB, together with the square of CB, is equal to the square of CD. Wherefore, if a straight line, &c. Q. E. D. PROP. VII. THEOR. If a straight line be divided into any two parts, the squares of the whole line, and of one of the parts are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. Let the straight line AB be divided into any two parts in the point C; the squares of AB, BC are equal to twice the rectangle AB, BC, together with the square of AC. Upon AB describe (46. 1.) the square ADEB, and construct the figure as in the preceding propositions : and because AG is equal (43. 1.) to GE, add to each of them CK; the whole AK is therefore equal to the whole CE; therefore AK, CE A С B are double of AK: but AK, CE are the gnomon AKF, together with the square CK; therefore the gnomon AKF, together G with the square CK, is double of AK: but H K twice the rectangle AB, BC is double of AK, for BK is equal (Cor. 4. 2.) to BC: therefore the gnomon AKF, together with the square CK, is equal to twice the rectangle AB, BC: to each of these equals add HF, D F E which is equal to the square of AC: therefore, the gnomon AKF, together with the squares CK, HF, is equal to twice the rectangle AB, BC, and the square of AC: but the gnomon AKF, together with the squares CK, HF, make up the whole figure ADEB and CK, which are the squares of AB and BC, therefore the squares of AB and BC are equal to twice the rectangle AB, BC, together with the square AC. Wherefore, if a straight line, &c. Q. E. D. 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, is equal to the square of the straight line made up of AB and BC together. Produce AB to D, so that BD be equal to CB, and upon AD de scribe the square AEFD; and construct two figures such as in the preceding. Because CB is equal to BD, and that CB is equal (34. 1.) 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 ĠK to KN, the rectangle CK is equal (36. 1.) to BN, and GR to RN; but CK is equal (43. 1.) to RN, because they are the complements of the parallelogram CO; therefore also BN is equal to GR; and the four rectangles BN, CK, GR, RN are therefore 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 (Cor. 4. 2.) to BK, that is, to CG ; C B and CB equal to GK, that (Cor. 4. 2,) A D is, to GP; therefore CG is equal to GP: G K and because CG is equal to GP, and PR M N to RO, the rectangle AG is equal to P R MP, and PL to RF: but MP is equal X O (43. 1.) to PL, because they are the complements of the parallelogram ML; wherefore AG is equal also to RF: therefore the four rectangles AG, MP, E H L PL, RF are equal to one another, and so F are quadruple of one of them AG. And it was demonstrated that the four CK, BN, GR, and RN are quadruple to 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, 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 add XH, which is equal (Cor. 4. 2.) to the square of AC: therefore four times the rectangle AB, BC, together with the square of AC, is equal to the gnomon 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. PROP. IX. THEOR. 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 at the point C into two equal, and at D into two unequal parts: the squares of AD, DB are together double of the squares of AC, CD. From the point C draw (11. 1.) CE at right angles to AB, and make it equal to AC or CB, and join EA, EB; through D draw (31. 1.) DF parallel to CE, and through F draw FG parallel to AB; and join AF: then, because AC is equal to CE, the angle EAC is equal (5. 1.) to the angle AEC; and because the angle ACE is a right angle, the two others AEC, EAC together make one right angle (32. 1.); and they are equal to one another; each of them therefore is half of a right angle. For the same A reason each of the angles CEB, EBC is half a right angle; and therefore the whole AEB is a right angle: and because G F the angle GEF is half a right angle, and EGF a right angle, for it is equal (29. 1.) to the interior and opposite angle ECB, the remaining angle EFG is half a right A С D B angle; therefore the angle GEF is equal to the angle EFG, and the side EG equal (6. 1.) to the side GF: again, because the angle at B is half a right angle, and FDB. half a right angle, for it is equal (29. 1.) to the interior and opposite angle ECB, the remaining angle BFD is half a right angle; therefore the angle at B is equal to the angle BFD, and the side DF to (6. 1.) the side DB: and because AC is equal to CE, the square of AC is equal to the square of CE; there. fore the squares of AC, CE are double of the square of AC: but the square of EA is equal (47. 1.) to the squares of AC, CE, because ACE is a right angle; therefore the square of EA 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 FG, 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 (34. 1.) 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; and the square of AF is equal (47. 1.) 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, DA 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 are double of the squares of AC, CD. From the point C draw (11. 1.) CE at right angles to AB: and make it equal to AC, or CB, and join AE, EB; through E draw (31. 1.) 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 (29. 1.) 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 do 'meet (12. Ax.) if produced far enough: therefore EB, FD shall meet, if produced towards B, D: let them meet in G, and join AG: then, because AC is equal to CE, the angle CEA is equal (5. 1.) to the angle EAC: and the angle ACE is a right angle: therefore each of the angles CEA, EAC is half a right angle (32. 1.): for the same reason, each of the angles CEB, EBC is half a right angle: therefore AEB is a right angle: and because EBC is half a right angle, DBG is also (15. 1.) half a right angle, for they are vertically opposite; but BDG is a right angle, because it is equal (29. 1.) to the alternate angle DCE; therefore the remaining angle DGB is half a right angle, and is therefore equal to the angle DBG; wherefore also the side BD is equal (6. 1.) to the side DG: again, because EGF is half a right angle, and that the angle at F is a right angle, because it is equal (34. 1.) to the E F opposite angle ECD, the remaining angle FEG is half a right angle, and equal to the angle EGF; wherefore also the side с B D to CA, the square of EC is equal to the square of CA; therefore the squares of EC, CA are dou G ble of the square of CA: but the square of EA is equal (47. 1.) to the squares of EC, CA; therefore the square of EA is double of 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 of GF, FE are double of the square of EF: but the square of EG is equal (47. 1.) 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 de monstrated, that the square of EA is double of the square of AC; therefore the squares of AE, EG are double of the squares of AC, CD: and the square of AG is equal (47. 1.) to the squares of AE, EG; therefore the square of AG is double of the squares of AC, CD: but the squares of AD, GD are equal (47. 1.) 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 AB describe (46. 1.) the square ABDC; bisect (10. 1.) AC in E, and join BE; produce CA to F, and make (3. 1.) EF equal to EB; and upon AF describe (46. 1.) the square FGHA; AB is divided in H, so that the rectangle AB, BH is equal to the square of AH. Produce GH to K; 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 (6. 2.) 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: and the squares of BA, AE are equal (47. 1.) to the square of F G 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 re н В maining rectangle CF, FA is equal to the A 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 E common part AK, and the remainder FH is equal to the remainder HD, and HD is the rectangle contained by AB, BH, for AB is equal to BD; and FH is the square of AH: C K D 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. |