PROPOSITION IV. THEOREM. If a straight line be divided into any two parts, the square on the whole line is equal to the squares on the two parts together with twice the rectangle contained by the parts. Ax. Ι. Let the st. line AB be divided into any two parts in C. Then must I. 46. Then :: BK=AH, .: BK=CB, Def. xxx. Now AE=sum of HG, CK, AF, FE, HG=sq. on AC, FE=rect. AC, CB, :: FG=AC and FK=CB. Q. E. D. Ex. In a triangle, whose vertical angle is a right angle, a straight line is drawn from the vertex perpendicular to the base. Shew that the rectangle, contained by the segments of the base, is equal to the square on the perpendicular. B PROPOSITION V. THEOREM. If a straight line be divided into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line. Let the st. line AB be divided equally in C and unequally in D. Then must rect. AD, DB together with sq. on CD=sq. on CB. I. 46. On CB describe the sq. CEFB. Draw DG || to CE, and from it cut off DH=DB. Draw HLK || to AD, and AK || to DH. I. 31. II. 31. Then rect. DF=rect. AL, Also LG=sq. on CD, :::BF=AC, and BD=CL. ::LH=CD, and HG=CD. Then rect. AD, DB together with sq. on CD =AH together with LG =sq. on CB. Q. E, D. PROPOSITION VI. THEOREM. If a straight line be bisected and produced to any point, the rectangle contained by the whole line thus produced and the part of it produced, together with the square on half the line bisected, is equal to the square on the straight line which is made up of the half and the part produced. Let the st. line AB be bisected in C and produced to D. I. 46. I. 31. Through H draw KLM || to AD I. 31. Through A draw AK || to CE. Now ::: BG=CD and BH=BD; .. HG=CB; Ax. 3. II. A. Then rect. AD, DB together with sq. on CB =sum of AM and LG *Sq. on CD Q. E, D. Note. We here give the proof of an important theorem. which is usually placed as a corollary to Proposition V. PROPOSITION B. THEOREM. The difference between the squares on any two straight lines is equal to the rectangle contained by the sum and difference of those lines. G F Let AC, CD be two st. lines, of which AC is the greater, and let them be placed so as to form one st. line AD. Produce AD to B, making CB=AC. and DB=the difference of the lines AC, CD. Then must difference between sqq. on AC, CD=rect. AD, DB. On OB describe the sq. CEFB. I. 46. Draw DG Iļ to CE, and from it cut off DH=DB. I. 31. Draw HLK || to AD, and AK || to DH. I. 31. Then rect. DF=rect. AL, :;:BF=AC, and BD=CL. Also LG=sq. on CD, :LH=CD, and HG=CD. =difference between sqq. on CB, CD Q. E. D. Ex. Shew that Propositions V. and VI. might be deduced from this Proposition. PROPOSITION VII. THEOREM. If a straight line be divided into any two parts, the squares on the whole line and on one of the parts are equal to twice the rectangle contained by the whole and that part together with the square on the other part. Let AB be divided into any two parts in C. I. 46. I. 31. Then sqq. on AB, BC=sum of AE and CK =sum of AK, HF, GE and CK Now AK=rect. AB, BC, :: BK=BC; CE=rect. AB, BC, :: BE=AB ; HF=sq. on AC. .:: sqq. on AB, BC=twice rect. AB, BC together with sq. on AC Q. E. D. Ex. If straight lines be drawn from G to B and from G to D, shew that BGD is a straight line. |