(36.) From one of the angular points of a given square to draw a line meeting one of the opposite sides, and the other produced, in such a manner, that the exterior triangle formed thereby may have a given ratio to the square. E D oc B G With the centre Let ABCD be the given square, and M: N the given ratio. From A to DC (produced if necessary) draw a line 40, such that M : M+N :: DC: AO. O and radius OA, describe a semicircle meeting DC produced in E and F. Join AF; which will be the line required. Join AE. Then M: M+N :: DC : A0 :: ABCD : the rectangle AO, AD. Now the triangle ADE is similar to ABG, and equal to it, since AB=AD; .. the trapezium AECG is equal to ABCD; and the rectangle AO, AD is equal to the triangle AEF, whence M: M+N:: AECG: AEF, .. M: N :: AECG: GCF N:: :: ABCD: GCF (37.) From a given point in the side produced, of a given rectangular parallelogram, to draw a line which shall cut the perpendicular sides and the other side produced, so that the trapezium cut off, which stands on the aforesaid side, may be to the triangle which stands upon the produced part of the opposite side, in a given ratio. Let AKCD be the given rectangle, and E the given point in the side CD produced. On EC describe a semicircle, and in it place E F K A H M DG EF=ED; join FC; and divide EC in G, so that EG: .. div. CDIM: ECM :: but ECM EDI :: EC2 - ED2 : EC2, .. ex æquo CDIM: BMK :: EC - EF2: BK :: FC: BK2 22 :: EG2 : GH2, by construction, :: EG: GC, i. e. in the given ratio. (38.) Through a given point, between two straight lines containing a given angle, to draw a line which shall cut off a triangle equal to a given figure. B I Ꮐ Let AB, AC be the lines containing the given angle BAC, and D the given point. Through D draw DE parallel to AC, and describe a parallelogram EG equal to the given figure. Draw GH perpendicular to AC, and equal to DE; and make HC =DF; join CD, and produce it to meet AB in B; CB is the line required. . H For the triangles EBD, DIF, GIC being similar, are to one another in the duplicate ratio of the sides DE, DF, GC; but the square of HC is equal to the squares of HG, GC; and.. the square of DF is equal to the squares of DE, GC; whence the triangle DIF is equal to the triangles DBE, GIC; .. the triangle ABC is equal to AEFG, i. e. to the given figure. (39.) Between two lines given in position, to draw a line equal to a given line, so that the triangle thus formed may be equal to a given rectilineal figure. K B Let AB, AC be the lines given in position, and DE the line whose magnitude is given. Bisect it in F, and on DF describe a rectangular parallelogram equal to the given figure. ment of a circle containing an angle equal to the angle at A, and cutting HG in I. Join DI, IE; and make AK ID, and AL-IE. Join KL; it is the line re quired. On DE describe a seg Since AK= ID, and AL= IE, and the angle at A= DIE, .. KL= DE, and the triangle AKL=IDE= HGFD=the given figure. (40.) From two given lines to cut off two others, so that the remainder of one may have to the part cut off from the other a given ratio; and the difference of the squares of the other remainder and part cut off from the first may be equal to a given square. Perpendicular to AB one of the given lines, draw BC equal to a side of the given square; and take AD to the other given line in the given ratio of the part remaining from the first to the part cut off from the BG and GF are equal to the parts to be cut off. For the difference between the squares of CG, GB is equal to the square of BC, i. e. to the given square; and AG : GF :: AD : AE, i. e. in the given ratio. (41.) From two given lines to cut off two others which shall have a given ratio, so that the difference of the squares of the remainders may be equal to a given square. Let AC be one of the two given lines. From C draw CD perpendicular to AC, and equal to a side of the given square. Take AE to the other given line in the given ratio of the parts to be cut off. Join ED, and produce it; and with the GE centre A, and radius equal to that other given line, describe a circle cutting ED in B. Join AB; and let it meet DF, which is parallel to AC, in F. Draw FG parallel to CD. CG and BF are the parts required to be cut off. For (DF =) CG : FB :: EA : AB, i. e. in the given ratio of the parts to be cut off, and the difference between the squares of FA and AG is equal to the square of GF, i. e. to the square of CD, or the given difference of the squares of the remainders. (42.) From two given lines to cut off two others so that the remainders may have a given ratio, and the sum of the squares of the parts cut off may be equal.to the square of a given line. F G Let AB be one of the given lines, and in it take AC to the other given line, in the given ratio of the remainders. From C draw CD perpendicular to AB, and equal to the second given line. Join AD, and draw CE parallel to AD; and with the centre B, and radius equal to the side of the given square, describe a circle, cutting CE in E. Draw EF parallel to DC. Then BG, GE will be equal to the parts to be cut off. Join BE. The squares of BG, GE are equal to the square of BE, i. e. to the given square; and AG: GF :: AC : CD, i. e. in the given ratio of the remainders. (43.) Two points being given in a given straight line; to determine a third such that the rectangles contained by its distances from each extremity and the given point adjacent to that extremity may be equal. and F. Bisect EF in G, and let fall the perpendicular GH; H is the point required. From G draw any lines GNK, GLM cutting the circles. Take O the centre of the circle ACE, and draw |