Questions for Examination on Book VI.
Questions for Examination on Book VI.
1.What is the subject-matter of Book VI.?Ans.Application of the theory of proportion.
2.What are similar rectilineal figures?
3.What do similar figures agree in?
4.How many conditions are necessary to define similar triangles?
5.How many to define similar rectilineal figures of more than three sides?
6.When is a figure said to be given in species?
7.When in magnitude?
8.When in position?
9.What is a mean proportional between two lines?
10.Define two mean proportionals.
11.What is the altitude of a rectilineal figure?
12.If two triangles have equal altitudes, how do their areas vary?
13.How do these areas vary if they have equal bases but unequal altitudes?
14.If both bases and altitudes differ, how do the areas vary?
15.When are two lines divided proportionally?
16.If in two lines divided proportionally a pair of homologous points coincide with their pointof intersection, what property holds for the lines joining the other pairs of homologouspoints?
17.Define reciprocal proportion.
18.If two triangles have equal areas, prove that their perpendiculars are reciprocallyproportional to the bases.
19.What is meant by figures inversely similar?
20.If two figures be inversely similar, how can they be changed into figures directlysimilar?
21.Give an example of two triangles inversely similar.Ans.If two lines passing through anypointOoutside a circle intersect it in pairs of pointsA,A′;B,B′, respectively, the trianglesOAB,OA′B′, are inversely similar.
22.What point is it round which a figure can be turned so as to bring its sides into positions ofparallelism with the sides of a similar rectilineal figure.Ans.The centre of similitude of the twofigures.
23.How many figures similar to a given rectilineal figure of sides can be described on a givenline?
24.How many centres of similitude can two regular polygons ofnsides each have?Ans.ncentres, which lie on a circle.
25.What are homothetic figures?
26.How do the areas of similar rectilineal figures vary?
27.What proposition isxix.a special case of?
28.Define Philo’s line.
29.How many centres of similitude have two circles?
Exercises on Book VI.
Exercises on Book VI.
1.If in a fixed triangle we draw a variable parallel to the base, the locus of the points ofintersection of the diagonals of the trapezium thus cut off from the triangle is the median thatbisects the base.
2.Find the locus of the point which divides in a given ratio the several lines drawn from a givenpoint to the circumference of a given circle.
3.Two linesAB,XY, are given in position:ABis divided inCin the ratiom:n, andparallelsAA′,BB′,CC′, are drawn in any direction meetingXYin the pointsA′,B′,C′;prove
(m + n)CC ′ = nAA′+mBB ′.
4.Three concurrent lines from the vertices of a triangleABCmeet the opposite sides inA′,B′,C′; prove
′ ′ ′ ′ ′ ′ AB .BC .CA =A B.B C.C A.
5.If a transversal meet the sides of a triangleABCin the pointsA′,B′,C′; prove
′ ′ ′ ′ ′ ′ AB .BC .CA = −A B.BC.C A.
6.If on a variable lineAC, drawn from a fixed pointAto any pointBin the circumference of agiven circle, a pointCbe taken such that the rectangleAB.ACis constant, the locus ofCis acircle.
7.IfDbe the middle point of the baseBCof a triangleABC,Ethe foot of the perpendicular,Lthe point where the bisector of the angleAmeetsBC,Hthe point of contact of the inscribedcircle withBC; proveDE.HL=HE.HD.
8.In the same case, ifKbe the point of contact withBCof the escribed circle, which touchesthe other sides produced,LH.BK=BD.LE.
9.IfR,r,r′,r′′,r′′′be the radii of the circumscribed, the inscribed, and the escribed circles ofa plane triangle,d,d′,d′′,d′′′the distances of the centre of the circumscribed circle from the centresof the others, thenR2=d2+ 2Rr=d′2−2Rr′, &c.
10.In the same case, 12R2=d2+d′2+d′′2+d′′′2.
11.Ifp′,p′′,p′′′denote the perpendiculars of a triangle, then
12.In a given triangle inscribe another of given form, and having one of its angles at a givenpoint in one of the sides of the original triangle.
13.If a triangle of given form move so that its three sides pass through three fixed points, thelocus of any point in its plane is a circle.
14.The angleAand the area of a triangleABCare given in magnitude: if the pointAbe fixedin position, and the pointBmove along a fixed line or circle, the locus of the pointCis acircle.
15.One of the vertices of a triangle of given form remains fixed; the locus of another is a rightline or circle; find the locus of the third.
16.Find the area of a triangle—(1) in terms of its medians; (2) in terms of its perpendiculars.
17.If two circles touch externally, their common tangent is a mean proportional between theirdiameters.
18.If there be given three parallel lines, and two fixed pointsA,B; then if the lines ofconnexion ofAandBto any variable point in one of the parallels intersect the other parallelsin the pointsCandD,EandF, respectively,CFandDEpass each through a fixedpoint.
19.If a system of circles pass through two fixed points, any two secants passing through one ofthe points are cut proportionally by the circles.
20.Find a pointOin the plane of a triangleABC, such that the diameters of thethree circles, about the trianglesOAB,OBC,OCA, may be in the ratios of three givenlines.
21.ABCDis a cyclic quadrilateral: the linesAB,AD, and the pointC, are given in position;find the locus of the point which dividesBDin a given ratio.
22.CA,CBare two tangents to a circle;BEis perpendicular toAD, the diameter throughA;prove thatCDbisectsBE.
23.If three lines from the vertices of a triangleABCto any interior pointOmeet the oppositesides in the pointsA′,B′,C′; prove
OA′ OB ′ OC′ AA′ + BB-′ + CC′-=1.
24.If three concurrent linesOA,OB,OCbe cut by two transversals in the two systems ofpointsA,B,C;A′,B′,C′, respectively: prove
-AB-.OC--=-BC-.OA- = CA-cOB--. A ′B′ OC′ B ′C ′OA′ C′A ′OB ′
25.The line joining the middle points of the diagonals of a quadrilateral circumscribed to acircle—
26.IfCD,CD′be the internal and external bisectors of the angleCof the triangleACB, thethree rectanglesAD.DB,AC.CB,AD.BD′are proportional to the squares ofAD,AC,AD′; andare—(1) in arithmetical progression if the difference of the base angles be equal to a right angle; (2)in geometrical progression if one base angle be right; (3) in harmonical progression if the sum of thebase angles be equal to a right angle.
27.If a variable circle touch two fixed circles, the chord of contact passes through a fixed pointon the line connecting the centres of the fixed circles.
Dem.—LetO,O′be the centres of the two fixed circles;O′′the centre of the variable circle;A,Bthe points of contact. LetABandOO′meet inC, and cut the fixed circles again in the pointsA′,B′respectively. JoinA′O,AO,BO′. ThenAO,BO′meet inO′′[III.xi.]. Now, because thetrianglesOAA′,O′′ABare isosceles, the angleO′′BA=O′′AB=OA′A. HenceOA′is paralleltoO′B; thereforeOC:O′C::OA′:O′B; that is, in a given ratio. HenceCis a givenpoint.
28.IfDD′be the common tangent to the two circles,DD′2=AB′.A′B.
29.IfRdenote the radius ofO′′andρ,ρ′, the radii ofO,O′,DD′2:AB2:: (R±ρ)(R±ρ′) :R2,the choice of sign depending on the nature of the contacts. This follows from 28.
30.If four circles be tangential to a fifth, and if we denote by12the common tangent to thefirst and second, &c., then
12.34+23.14 =13.24.
31.The inscribed and escribed circles of any triangle are all touched by its nine-pointscircle.
32.The four triangles which are determined by four points, taken three by three, are such thattheir nine-points circles have one common point.
33.Ifa,b,c,ddenote the four sides, andD,D′the diagonals of a quadrilateral; prove that thesides of the triangle, formed by joining the feet of the perpendiculars from any of its angular pointson the sides of the triangle formed by the three remaining points, are proportional to the threerectanglesac,bd,DD′.
34.Prove the converse of Ptolemy’s theorem(seexvii., Ex.13).
35.Describe a circle which shall—(1) pass through a given point, and touch two given circles;(2)touch three given circles.
36.If a variable circle touch two fixed circles, the tangent to it from their centre of similitude,through which the chord of contact passes (27), is of constant length.
37.If the linesAD,BD′(seefig., Ex.27) be produced, they meet in a point on thecircumference ofO′′, and the lineO′′Pis perpendicular toDD′.
38.IfA,Bbe two fixed points on two lines given in position, andA′,B′two variable points,such that the ratioAA′:BB′is constant, the locus of the point dividingA′B′in a given ratio is aright line.
39.If a lineEFdivide proportionally two opposite sides of a quadrilateral, and a lineGHtheother sides, each of these is divided by the other in the same ratio as the sides which determinethem.
40.In a given circle inscribe a triangle, such that the triangle whose angular points are the feetof the perpendiculars from the extremities of the base on the bisector of the verticalangle, and the foot of the perpendicular from the vertical angle on the base, may be amaximum.
41.In a circle, the point of intersection of the diagonals of any inscribed quadrilateral coincideswith the point of intersection of the diagonals of the circumscribed quadrilateral, whose sides touchthe circle at the angular points of the inscribed quadrilateral.
42.Through two given points describe a circle whose common chord with another given circlemay be parallel to a given line, or pass through a given point.
43.Being given the centre of a circle, describe it so as to cut the legs of a given angle along achord parallel to a given line.
44.If concurrent lines drawn from the angles of a polygon of an odd number of sides divide theopposite sides each into two segments, the product of one set of alternate segments is equal to theproduct of the other set.
45.If a triangle be described about a circle, the lines from the points of contact of its sides withthe circle to the opposite angular points are concurrent.
46.If a triangle be inscribed in a circle, the tangents to the circle at its three angular pointsmeet the three opposite sides at three collinear points.
47.The external bisectors of the angles of a triangle meet the opposite sides in three collinearpoints.
48.Describe a circle touching a given line at a given point, and cutting a given circle at a givenangle.
Def.—The centre of mean position of any number of pointsA,B,C,D, &c.,is a point whichmay be found as follows:—Bisect the line joining any two pointsA,B, inG. JoinGto a third pointC; divideGCinH, so thatGH=13GC. JoinHto a fourth pointD, and divideHDinK, so thatHK=14HD, and so on. The last point found will be the centre of mean position of the givenpoints.
49.The centre of mean position of the angular points of a regular polygon is the centre of figureof the polygon.
50.The sum of the perpendiculars let fall from any system of pointsA,B,C,D, &c., whosenumber isnon any lineL, is equal tontimes the perpendicular from the centre of mean position onL.
51.The sum of the squares of lines drawn from any system of pointsA,B,C,D, &c., to anypointP, exceeds the sum of the squares of lines from the same points to their centre of meanposition,O, bynOP2.
52.If a point be taken within a triangle, so as to be the centre of mean position of the feet ofthe perpendiculars drawn from it to the sides of the triangle, the sum of the squares of theperpendiculars is a minimum.
53.Construct a quadrilateral, being given two opposite angles, the diagonals, and the anglebetween the diagonals.
54.A circle rolls inside another of double its diameter; find the locus of a fixed point in itscircumference.
55.Two points,C,D, in the circumference of a given circle are on the same side of a givendiameter; find a pointPin the circumference at the other side of the given diameter,AB, such thatPC,PDmay cutABat equal distances from the centre.
56.If the sides of any polygon be cut by a transversal, the product of one set of alternatesegments is equal to the product of the remaining set.
57.A transversal being drawn cutting the sides of a triangle, the lines from the angles of thetriangle to the middle points of the segments of the transversal intercepted by those angles meet theopposite sides in collinear points.
58.If lines be drawn from any pointPto the angles of a triangle, the perpendiculars atPtothese lines meet the opposite sides of the triangle in three collinear points.
59.Divide a given semicircle into two parts by a perpendicular to the diameter, so that the radiiof the circles inscribed in them may have a given ratio.
60.From a point within a triangle perpendiculars are let fall on the sides; find the locus of thepoint, when the sum of the squares of the lines joining the feet of the perpendiculars isgiven.
61.If a circle make given intercepts on two fixed lines, the rectangle contained bythe perpendiculars from its centre on the bisectors of the angle formed by the lines isgiven.
62.If the base and the difference of the base angles of a triangle be given, the rectanglecontained by the perpendiculars from the vertex on two lines through the middle point of the base,parallel to the internal and external bisectors of the vertical angle, is constant.
63.The rectangle contained by the perpendiculars from the extremities of the base of atriangle, on the internal bisector of the vertical angle, is equal to the rectangle contained bythe external bisector and the perpendicular from the middle of the base on the internalbisector.
64.State and prove the corresponding theorem for perpendiculars on the externalbisector.
65.IfR,R′denote the radii of the circles inscribed in the triangles into which a right-angledtriangle is divided by the perpendicular from the right angle on the hypotenuse; then, ifcbe thehypotenuse, andsthe semiperimeter,R2+R′2= (s−c)2.
66.IfA,B,C,Dbe four collinear points, find a pointOin the same line with them such thatOA.OD=OB.OC.
67.The four sides of a cyclic quadrilateral are given; construct it.
68.Being given two circles, find the locus of a point such that tangents from it to the circlesmay have a given ratio.
69.If four pointsA,B,C,Dbe collinear, find the locus of the pointPat whichABandCDsubtend equal angles.
70.If a circle touch internally two sides,CA,CB, of a triangle and its circumscribed circle, thedistance fromCto the point of contact on either side is a fourth proportional to the semiperimeter,andCA,CB.
71.State and prove the corresponding theorem for a circle touching the circumscribed circleexternally and two sides produced.