Book Il. PRO P. IX. THEOR. Fa straight line be divided into two equal, and also in to 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 fe&ion. Let the straight line AB be divided at the point C into two equal, and at Ď into two unequal parts : The squares of AD, DB are together double of the squares of AC, CD. From the point C draw. CE at right angles to AB, and a 11. I. make it equal to AC or CB, and join EA, EB; through D draw ► DF parallel to CE, and through F draw FG parallel to AB; b 31. I. and join AF: Then, because AC is equal to CE, the angle EAC is equal to the angle AEC; and because the angle c 5.1. ACE is a right angle, the two others AEC, EAC together make one right angled; and they are equal to one another; d 32. 1. each of them therefore is half of a right angle. For the same reason each of the angles CEB, EBC is half a night angle; and G E therefore the whole AEB is a right angle: And becaute theangle GEF is half a right angle, and EGF a right angle, for it is e- A C D B qual' to the interior and oppo € 29.1. ute angle ECB, the remaining angle EFG is balf a right angle; therefore the angle GEF is equal to the angle EFG, and the hde EG equal to the Gide GF: Again, because the angle at B 8 6.1. is half a right angle, and FDB a right angle, for it is equal to the interior and opposte angle ECB, the remaining angle BFD is half a righe angle; therefore the angle at B is equal to the angle BFD, and the lide DF tof the fide DB : And be. cause AČ 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 EA is equals to the 8 49. I. 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 Iquare of GF; therefore the squares of EG, GF are double of the Book II. the square of GF; but the square of EF is equal to the squares W of EG, GF; therefore the square of EF is double of the square h 34. I. GF: And GF is equalh 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 iquare of AF is i 47.1. equal i 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 fquares 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. PRO P. X. THÉ O R. a straight line be bisected, and produced to any point, IT 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 produces. 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 a CE at right angles to AB: And make it equal to AC or CB, and join AE, EB; through E draw 1 31. 1. B 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 CŁF, 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 fide less than two right angles, do meet d if produced far enough: Therefore EB, FD fhall meet, if produced, toward B, D: Let them meet in G, and join AG: Then, because 65. I. - AC is equal to CE, the angle CEA is equal to the angle EAC; and the angle ACE is a right angle; therefore each of the $ 32. I. angles CEA, LAC is half a right angler: For the same reason, each 12. Ax. each of the angles CEB, EBC is half a right angle; therefore Book II. AEB is a right angle: And because EBC is half a right angle, DBG is alsof half a right angle, for they are vertically oppo- f 15. I. Gre; but BDG is a right angle, because it is equal to the al-c 29. I. ternate ang!e DCE; therefore the remaining angle DGB is half a right angle, and is therefore equal to the angle DBG; wherefore also the fide BD is equals to the Gide DG: Again, 8 6. I. because EGF is half a right angle, and that E F the angle at F is a right angle, because it is e. qual to the opposite h 34. 1. angle ECD, the remaining angle FEG is half a с A BD right angle, and equal to the angle EGF, wherefore also the fide G GF is equal to the fide FE. And because EC is equal to CA, the square of EC is equal to the square of CA ; therefore the squares of EC, CA are double of the square of CA: But the square of EA is equal i to the squares of EC, CA; there. i 49. I, fore 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 dou. ble of the square of EF: But the square of EG is equal i 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 AE, EG are double of the squares of AC, CD: And the square of AG is equal i to the squares of AE, EG; therefore the square of AG is double of the squares of AC, CD: But the squares of AD, DG are equal i to the square of AG; therefore the squares of AD, DG are double of the squares of AC, CD : Bui DG is equal to DB; therefore ihe squares of AD, DB are double of the squares of AC, CD: Wherefore, if a straight line, &c. QE D. PROP. Book II, PRO P. XI. PRO B. no 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. l'pon AB describe the square ABDC; bisectAC in E, and join BE; produce CA to F, and make EF equal to EB; and upon AF describe 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 cqual d to the fquare of EF : But EF is equal to EB; therefore the rectangle CF, FA, toge. ther with the square of AE, is equal to the square of EB: And the squares of BA, AE are equal to the F G H B E K D PROP. Book II. PRO P. XII. THE O R. O [N obtuse angled triangles, if a perpendicular be drawn from any of the acute angles to the opposite side produced, the square of the side fubtending the obtuse angle is greater than the squares of 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 • perpen- a 12. I. dicular to BC produced : The square of AB is greater than the squares of AC, CB by twice the rectangle BC, CD. Because the straight line BD is divided into two parts in the point C, the square of BD is equal 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 BD, DA are equal to the squares of BC, CD, DA, and twice the rectangle BC, CD: But the square of BA is equal to the squares of BD, DA, be C 47.5. cause the angle at D is a right B angle; and the square of CA is e D quale 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 BA is greater than the squares of BC, CA, by twice the rectangle BC, CD. Therefore, in obtufe angled triangles, &c. Q. E. D. С PROP |