72.Pascal’s Theorem.—If the opposite sides of an irregular hexagonABCDEFinscribed in a circle be produced till they meet, the three points of intersectionG,H,Iarecollinear.
Dem.—JoinAD. Describe a circle about the triangleADI, cutting the linesAF,CDproduced,if necessary, inKandL. JoinIK,KL,LI. Now, the anglesKLG,FCGare each [III.xxi.] equal tothe angleGAD. Hence they are equal. ThereforeKLis parallel toCF. Similarly,LIisparallel toCH, andKItoFH; hence the trianglesKLI,FCHare homothetic. Hence thelines joining corresponding vertices are concurrent. Therefore the pointsI,H,Garecollinear.
73.If two sides of a triangle circumscribed to a given circle be given in position, but the thirdside variable, the circle described about the triangle touches a fixed circle.
74.If two sides of a triangle be given in position, and if the area be given in magnitude, twopoints can be found, at each of which the base subtends a constant angle.
75.Ifa,b,c,ddenote the sides of a cyclic quadrilateral, andsits semiperimeter, prove its area=∘ (s−-a)(s−-b)(s− c)(s-− d).
76.If three concurrent lines from the angles of a triangleABCmeet the opposite side inthe pointsA′,B′,C′, and the pointsA′,B′,C′be joined, forming a second triangleA′B′C′,
′ ′ ′ ′ ′ ′ △ ABC :△ A B C ::AB.BC.CA :2AB .BC .CA .
77.In the same case the diameter of the circle circumscribed about the triangleABC=AB′.BC′.CA′divided by the area ofA′B′C′.
78.If a quadrilateral be inscribed in one circle, and circumscribed to another, the square of itsarea is equal to the product of its four sides.
79.If on the sidesAB,ACof a triangleABCwe take two pointsD,E, and on their line ofconnexionF, such that
BD-= AE-= DF-; AD CE EF
prove the triangleBFC= 2ADE.
80.If through the middle points of each of the two diagonals of a quadrilateral we draw aparallel to the other, the lines drawn from their points of intersection to the middle points of thesides divide the quadrilateral into four equal parts.
81.CE,DFare perpendiculars to the diameter of a semicircle, and two circles are describedtouchingCE,DE, and the semicircle, one internally and the other externally; the rectanglecontained by the perpendiculars from their centres onABis equal toCE.DF.
82.If lines be drawn from any point in the circumference of a circle to the angular points of anyinscribed regular polygon of an odd number of sides, the sums of the alternate lines areequal.
83.If at the extremities of a chord drawn through a given point within a given circle tangentsbe drawn, the sum of the reciprocals of the perpendiculars from the point upon the tangents isconstant.
84.If a cyclic quadrilateral be such that three of its sides pass through three fixed collinearpoints, the fourth side passes through a fourth fixed point, collinear with the three givenones.
85.If all the sides of a polygon be parallel to given lines, and if the loci of all the angles but onebe right lines, the locus of the remaining angle is also a right line.
86.If the vertical angle and the bisector of the vertical angle be given, the sum of thereciprocals of the containing sides is constant.
87.IfP,P′denote the areas of two regular polygons of any common number of sides, inscribedand circumscribed to a circle, and Π, Π′the areas of the corresponding polygons of double thenumber of sides; prove Π is a geometric mean betweenPandP′, and Π′a harmonic mean betweenΠ andP′.
88.The difference of the areas of the triangles formed by joining the centres of the circlesdescribed about the equilateral triangles constructed—(1) outwards; (2) inwards—on the sides ofany triangle, is equal to the area of that triangle.
89.In the same case, the sum of the squares of the sides of the two new triangles is equal to thesum of the squares of the sides of the original triangle.
90.IfR,rdenote the radii of the circumscribed and inscribed circles to a regular polygon of anynumber of sides,R′,r′, corresponding radii to a regular polygon of the same area, and double thenumber of sides; prove
√--- ∘------- R′ = Rr, and r′ = r(R-+r). 2
91.If the altitude of a triangle be equal to its base, the sum of the distances of theorthocentre from the base and from the middle point of the base is equal to half thebase.
92.In any triangle, the radius of the circumscribed circle is to the radius of the circle which isthe locus of the vertex, when the base and the ratio of the sides are given, as the difference of thesquares of the sides is to four times the area.
93.Given the area of a parallelogram, one of its angles, and the difference between its diagonals;construct the parallelogram.
94.If a variable circle touch two equal circles, one internally and the other externally, andperpendiculars be let fall from its centre on the transverse tangents to these circles, the rectangle ofthe intercepts between the feet of these perpendiculars and the intersection of the tangents isconstant.
95.Given the base of a triangle, the vertical angle, and the point in the base whose distancefrom the vertex is equal half the sum of the sides; construct the triangle.
96.If the middle point of the baseBCof an isosceles triangleABCbe the centre of a circletouching the equal sides, prove that any variable tangent to the circle will cut the sides in pointsD,E, such that the rectangleBD.CEwill be constant.
97.Inscribe in a given circle a trapezium, the sum of whose opposite parallel sides is given, andwhose area is given.
98.Inscribe in a given circle a polygon all whose sides pass through given points.
99.If two circlesX,Ybe so related that a triangle may be inscribed inXand circumscribedaboutY, an infinite number of such triangles can be constructed.
100.In the same case, the circle inscribed in the triangle formed by joining the points of contactonYtouches a given circle.
101.And the circle described about the triangle formed by drawing tangents toX, at theangular points of the inscribed triangle, touches a given circle.
102.Find a point, the sum of whose distances from three given points may be aminimum.
103.A line drawn through the intersection of two tangents to a circle is divided harmonically bythe circle and the chord of contact.
104.To construct a quadrilateral similar to a given one whose four sides shall pass through fourgiven points.
105.To construct a quadrilateral, similar to a given one, whose four vertices shall lie on fourgiven lines.
106.Given the base of a triangle, the difference of the base angles, and the rectangle of thesides; construct the triangle.
107.ABCDis a square, the sideCDis bisected inE, and the lineEFdrawn, making the angleAEF=EAB; prove thatEFdivides the sideBCin the ratio of 2 : 1.
108.If any chord be drawn through a fixed point on a diameter of a circle, and its extremitiesjoined to either end of the diameter, the joining lines cut off, on the tangent at the other end,portions whose rectangle is constant.
109.If two circles touch, and through their point of contact two secants be drawn at rightangles to each other, cutting the circles respectively in the pointsA,A′;B,B′; thenAA′2+BB′2isconstant.
110.If two secants at right angles to each other, passing through one of the points ofintersection of two circles, cut the circles again, and the line through their centres in the two systemsof pointsa,b,c;a′,b′,c′respectively, thenab:bc::a′b′:b′c′.
111.Two circles described to touch an ordinate of a semicircle, the semicircle itself, and thesemicircles on the segments of the diameter, are equal to one another.
112.If a chord of a given circle subtend a right angle at a given point, the locus of theintersection of the tangents at its extremities is a circle.
113.The rectangle contained by the segments of the base of a triangle, made by the point ofcontact of the inscribed circle, is equal to the rectangle contained by the perpendiculars from theextremities of the base on the bisector of the vertical angle.
114.IfObe the centre of the inscribed circle of the triangle prove
OA2-+ OB2-+ OC2-= 1. bc ca ab
115.State and prove the corresponding theorems for the centres of the escribed circles.
116.Four pointsA,B,C,Dare collinear; find a pointPat which the segmentsAB,BC,CDsubtend equal angles.
117.The product of the bisectors of the three angles of a triangle whose sides area,b,c,is
---8abc.s.area---- (a+ b)(b+c)(c+ a).
118.In the same case the product of the alternate segments of the sides made by the bisectorsof the angles is
-----a2b2c2-----. (a+ b)(b+c)(c+ a)
119.If three of the six points in which a circle meets the sides of any triangle be such, that thelines joining them to the opposite vertices are concurrent, the same property is true of the threeremaining points.
120.If a triangleA′B′C′be inscribed in anotherABC, prove
′ ′ ′ ′ ′ ′ AB .BC .CA + AB.B C.CA
is equal twice the triangleA′B′C′multiplied by the diameter of the circleABC.
121.Construct a polygon of an odd number of sides, being given that the sides taken in orderare divided in given ratios by fixed points.
122.If the external diagonal of a quadrilateral inscribed in a given circle be a chord of anothergiven circle, the locus of its middle point is a circle.
123.If a chord of one circle be a tangent to another, the line connecting the middle point ofeach arc which it cuts off on the first, to its point of contact with the second, passes through a givenpoint.
124.From a pointPin the plane of a given polygon perpendiculars are let fall on its sides; ifthe area of the polygon formed by joining the feet of the perpendiculars be given, the locus ofPis acircle.