Dem.—1. LetPApass through the centreO. JoinOT. Then becauseABis bisected inOand divided externally inP, the rectangleAP.BP+OB2is equal toOP2[II.vi.]. But sincePTis a tangent, andOTdrawn from the centre to the point of contact, the angleOTPis right [xviii.]. HenceOT2+PT2is equal toOP2.
2. IfABdoes not pass through the centreO, let fall the perpendicularOConAB. JoinOT,OB,OP. Then becauseOC, a line through the centre, cutsAB, which does not pass through the centre at right angles, it bisects it [iii.]. Hence, sinceABis bisected inCand divided externally inP, the rectangle
Hence, adding, sinceCB2+OC2=OB2[I.xlvii.], andCP2+OC2=OP2, we get
and rejecting the equalsOB2andOT2,we have the rectangle
2 AP .BP = P T .
The two Propositionsxxxv.,xxxvi., may be included in one enunciation, as follows:—TherectangleAP.BPcontained by the segments of any chord of a given circle passing through a fixedpointP, either within or without the circle, is constant.For letObe the centre: joinOA,OB,OP.ThenOABis an isosceles triangle, andOPis a line drawn from its vertex to a pointPin the base,or base produced. Then the rectangleAP.BPis equal to the difference of the squares ofOBandOP, and is therefore constant.
Cor. 1.—If two linesAB,CDproduced meet inP, and if the rectangleAP.BP=CP.DP, the pointsA,B,C,Dare concyclic (comparexxxv.,Cor.2).
Cor.2.—Tangents to two circles from any point in their common chord are equal (comparexvii., Ex. 6).
Cor.3.—The common chords of any three intersecting circles are concurrent (comparexvii., Ex. 7).
Exercise.
Exercise.
If from the vertexAof a△ABC,ADbe drawn, meetingCBproduced inD, and making theangleBAD=ACB, proveDB.DC=DA2.
PROP.XXXVII.—Theorem.
PROP.XXXVII.—Theorem.
If the rectangle(AP.BP)contained by the segments of a secant, drawn fromany point(P)without a circle, be equal to the square of a line(PT)drawnfrom the same point to meet the circle, the line which meets the circle is atangent.
Dem.—FromPdrawPQtouching the circle [xvii.]. LetObe the centre. JoinOP,OQ,OT. Now the rectangleAP.BPis equal to the square onPT(hyp.), and equal to the square onPQ[xxxvi.]. HencePT2is equal toPQ2, and thereforePTis equal toPQ. Again, the trianglesOTP,OQPhave the sideOTequalOQ,TPequalQP, and the baseOPcommon; hence [I.viii.] the angleOTPis equal toOQP; butOQPis a right angle, sincePQis a tangent [xviii.]; henceOTPis right,and therefore[xvi.]PTis atangent.
Exercises.
Exercises.
1.Describe a circle passing through two given points, and fulfilling either of the followingconditions: 1, touching a given line; 2, touching a given circle.
2.Describe a circle through a given point, and touching two given lines; or touching a given fileand a given circle.
3.Describe a circle passing through a given point, having its centre on a given line and touchinga given circle.
4.Describe a circle through two given points, and intercepting a given arc on a givencircle.
5.A,B,C,Dare four collinear points, andEFis a common tangent to the circles describeduponAB,CDas diameters: prove that the trianglesAEB,CFDare equiangular.
6.The diameter of the circle inscribed in a right-angled triangle is equal to half the sum of thediameters of the circles touching the hypotenuse, the perpendicular from the right angle of thehypotenuse, and the circle described about the right-angled triangle.
Questions for Examination on Book III.
Questions for Examination on Book III.
1.What is the subject-matter of Book III.?
2.Define equal circles.
3.What is the difference between a chord and a secant?
4.When does a secant become a tangent?
5.What is the difference between a segment of a circle and a sector?
6.What is meant by an angle in a segment?
7.If an arc of a circle be one-sixth of the whole circumference, what is the magnitude of theangle in it?
8.What are linear segments?
9.What is meant by an angle standing on a segment?
10.What are concyclic points?
11.What is a cyclic quadrilateral?
12.How many intersections can a line and a circle have?
13.What does the line become when the points of intersection become consecutive?
14.How many points of intersection can two circles have?
15.What is the reason that if two circles touch they cannot have any other commonpoint?
16.Give one enunciation that will include Propositionsxi.,xii.of BookIII.
17.What Proposition is this a limiting case of?
18.Explain the extended meaning of the word angle.
19.What is Euclid’s limit of an angle?
20.State the relations between Propositionsxvi.,xviii.,xix.
21.What Propositions are these limiting cases of?
22.How many common tangents can two circles have?
23.What is the magnitude of the rectangle of the segments of a chord drawn through a point3.65 metres distant from the centre of a circle whose radius is 4.25 metres?
24.The radii of two circles are 4.25 and 1.75 feet respectively, and the distance between theircentres 6.5 feet; find the lengths of their direct and their transverse common tangents.
25.If a point behfeet outside the circumference of a circle whose diameter is 7920 miles, provethat the length of the tangent drawn from it to the circle is∘ --- 3h 2miles.
26.Two parallel chords of a circle are 12 perches and 16 perches respectively, and their distanceasunder is 2 perches; find the length of the diameter.
27.What is the locus of the centres of all circles touching a given circle in a givenpoint?
28.What is the condition that must be fulfilled that four points may be concyclic?
29.If the angle in a segment of a circle be a right angle and a-half, what part of the wholecircumference is it?
30.Mention the converse Propositions of BookIII. which are proved directly.
31. What is the locus of the middle points of equal chords in a circle?
32.The radii of two circles are 6 and 8, and the distance between their centres 10; find thelength of their common chord.
33.If a figure of any even number of sides be inscribed in a circle, prove that the sum of one setof alternate angles is equal to the sum of the remaining angles.
Exercises on Book III.
Exercises on Book III.
1.If two chords of a circle intersect at right angles, the sum of the squares on their segments isequal to the square on the diameter.
2.If a chord of a given circle subtend a right angle at a fixed point, the rectangle of theperpendiculars on it from the fixed point and from the centre of the given circle is constant. Also thesum of the squares of perpendiculars on it from two other fixed points (which may be found) isconstant.
3.If through either of the points of intersection of two equal circles any line be drawn meetingthem again in two points, these points are equally distant from the other intersection of thecircles.
4.Draw a tangent to a given circle so that the triangle formed by it and two fixed tangents tothe circle shall be—1, a maximum; 2, a minimum.
5.If through the points of intersectionA,Bof two circles any two linesACD,BEFbe drawnparallel to each other, and meeting the circles again inC,D,E,F; thenCD=EF.
6.In every triangle the bisector of the greatest angle is the least of the three bisectors of theangles.
7.The circles whose diameters are the four sides of any cyclic quadrilateral intersect again infour concyclic points.
8.The four angular points of a cyclic quadrilateral determine four triangles whose orthocentres(the intersections of their perpendiculars) form an equal quadrilateral.
9.If through one of the points of intersection of two circles we draw two commonchords, the lines joining the extremities of these chords make a given angle with eachother.
10.The square on the perpendicular from any point in the circumference of a circle, on thechord of contact of two tangents, is equal to the rectangle of the perpendiculars from the same pointon the tangents.
11.Find a point in the circumference of a given circle, the sum of the squares on whosedistances from two given points may be a maximum or a minimum.
12.Four circles are described on the sides of a quadrilateral as diameters. The commonchord of any two on adjacent sides is parallel to the common chord of the remainingtwo.
13.The rectangle contained by the perpendiculars from any point in a circle, on the diagonals ofan inscribed quadrilateral, is equal to the rectangle contained by the perpendiculars from the samepoint on either pair of opposite sides.
14.The rectangle contained by the sides of a triangle is greater than the square on theinternal bisector of the vertical angle, by the rectangle contained by the segments of thebase.
15.If throughA, one of the points of intersection of two circles, we draw any lineABC, cutting the circles again inBandC, the tangents atBandCintersect at a givenangle.
16.If a chord of a given circle pass through a given point, the locus of the intersection oftangents at its extremities is a right line.
17.The rectangle contained by the distances of the point where the internal bisector of thevertical angle meets the base, and the point where the perpendicular from the vertex meets itfrom the middle point of the base, is equal to the square on half the difference of thesides.
18.State and prove the Proposition analogous to 17 for the external bisector of the verticalangle.
19.The square on the external diagonal of a cyclic quadrilateral is equal to the sum of thesquares on the tangents from its extremities to the circumscribed circle.
20.If a variable circle touch a given circle and a given line, the chord of contact passes througha given point.
21.IfA,B,Cbe three points in the circumference of a circle, andD,Ethe middle points ofthe arcsAB,AC; then if the lineDEintersect the chordsAB,ACin the pointsF,G,AFis equaltoAG.
22.Given two circles,O,O′; then if any secant cutOin the pointsB,C, andO′in the pointsB′,C′, and another secant cuts them in the pointsD,E;D′,E′respectively; the four chordsBD,CE,B′D′,C′E′form a cyclic quadrilateral.
23.If a cyclic quadrilateral be such that a circle can be inscribed in it, the lines joining thepoints of contact are perpendicular to each other.
24.If through the point of intersection of the diagonals of a cyclic quadrilateral the minimumchord be drawn, that point will bisect the part of the chord between the opposite sides of thequadrilateral.
25.Given the base of a triangle, the vertical angle, and either the internal or the externalbisector at the vertical angle; construct it.
26.If through the middle pointAof a given arcBACwe draw any chordAD, cuttingBCinE,the rectangleAD.AEis constant.
27.The four circles circumscribing the four triangles formed by any four lines pass through acommon point.
28.IfX,Y,Zbe any three points on the three sides of a triangleABC, the three circles aboutthe trianglesY AZ,ZBX,XCYpass through a common point.
29.If the position of the common point in the last question be given, the three angles of thetriangleXY Zare given, and conversely.
30.Place a given triangle so that its three sides shall pass through three given points.
31.Place a given triangle so that its three vertices shall lie on three given lines.
32.Construct the greatest triangle equiangular to a given one whose sides shall pass throughthree given points.
33.Construct the least triangle equiangular to a given one whose vertices shall lie on three givenlines.
34.Construct the greatest triangle equiangular to a given one whose sides shall touch threegiven circles.
35.If two sides of a given triangle pass through fixed points, the third touches a fixedcircle.
36.If two sides of a given triangle touch fixed circles, the third touches a fixed circle.
37.Construct an equilateral triangle having its vertex at a given point, and the extremities ofits base on a given circle.
38.Construct an equilateral triangle having its vertex at a given point, and the extremities ofits base on two given circles.
39.Place a given triangle so that its three sides shall touch three given circles.
40.Circumscribe a square about a given quadrilateral.
41.Inscribe a square in a given quadrilateral.
42.Describe circles—(1) orthogonal(cutting at right angles) to a given circle and passingthrough two given points; (2) orthogonal to two others, and passing through a given point; (3)orthogonal to three others.
43.If from the extremities of a diameterABof a semicircle two chordsAD,BEbe drawn,meeting inC,AC.AD+BC.BE=AB2.
44.IfABCDbe a cyclic quadrilateral, and if we describe any circle passing through the pointsAandB, another throughBandC, a third throughCandD, and a fourth throughDandA; thesecircles intersect successively in four other pointsE,F,G,H, forming another cyclicquadrilateral.
45.IfABCbe an equilateral triangle, what is the locus of the pointM, ifMA=MB+MC?
46.In a triangle, given the sum or the difference of two sides and the angle formed by thesesides both in magnitude and position, the locus of the centre of the circumscribed circle is a rightline.
47.Describe a circle—(1) through two given points which shall bisect the circumference of agiven circle; (2) through one given point which shall bisect the circumference of two givencircles.
48.Find the locus of the centre of a circle which bisects the circumferences of two givencircles.
49.Describe a circle which shall bisect the circumferences of three given circles.
50.ABis a diameter of a circle;AC,ADare two chords meeting the tangent atBin the pointsE,Frespectively: prove that the pointsC,D,E,Fare concyclic.
51.CDis a perpendicular from any pointCin a semicircle on the diameterAB;EFGis acircle touchingDBinE,CDinF, and the semicircle inG; prove—(1) that the pointsA,F,Garecollinear; (2) thatAC=AE.
52.Being given an obtuse-angled triangle, draw from the obtuse angle to the opposite side a linewhose square shall be equal to the rectangle contained by the segments into which it divides theopposite side.
53.Ois a point outside a circle whose centre isE; two perpendicular lines passing throughOintercept chordsAB,CDon the circle; thenAB2+CD2+ 4OE2= 8R2.
54.The sum of the squares on the sides of a triangle is equal to twice the sum of the rectanglescontained by each perpendicular and the portion of it comprised between the corresponding vertexand the orthocentre; also equal to 12R2minus the sum of the squares of the distances of theorthocentre from the vertices.
55.If two circles touch inC, and ifDbe any point outside the circles at which theirradii throughCsubtend equal angles, ifDE,DFbe tangent fromD,DE.DF=DC2.