Ex. 55. O is the centre and OA the radius of a circle. Another circle is described on OA as diameter. OPQ cuts the smaller circle in P and the larger in Q. Prove that, if the circle on OA as diameter roll on the inside of the larger circle, the point P will always fall on the point Q or on the point at which QO produced cuts the larger circle. Ex. 56. In fig. 110, AB is a tangent; OD DA=AB. BD cuts the circumference at E. Prove that arc AE is and arc EF 24 of the circumference. Ex. 57. ABC and ADEFG are respectively an equilateral triangle and a regular pentagon inscribed in a circle. What fraction of the circumference is the arc BD? Fig. 110. B Loci. Ex. 58. The base BC of a triangle is fixed and the vertical angle A is constant. Find the locus of the feet of the perpendiculars from B and C on AC and AB. Ex. 59. A straight line AB of constant length moves with A and B on two perpendicular straight lines. Prove that the locus of the mid-point of AB is a circle. Ex. 60. A straight line AB of constant length moves with its extremities on two lines OL, OM at right angles to one another. If C be the remaining angular point of the rectangle whose sides are OA, OB, find the locus of C. Ex. 61. Prove that the locus of the mid-points of chords of a circle which are drawn through a fixed point is a circle. Ex. 62. P is a variable point on an arc AB. AP is produced to Q so that PQ=PB. Prove that the locus of Q is a circular arc. Ex. 63. The base of a triangle is fixed and the vertical angle is of constant magnitude. Find the locus of the intersection of the perpendiculars from the extremities of the base on to the opposite sides. A Ex. 64. Prove that BIC = 90° + where is the inscribed centre of 2' Δ ΑΒΕ. Hence find the locus of the inscribed centre of a triangle whose base and vertical angle are given. Ex. 65. The base of a triangle is fixed and the vertical angle is constant. Find the locus of the mid-point of the line from the vertex to the middle point of the base. [Draw parallels.] Ex. 66. Given the middle points of two adjacent sides of a rectangle, prove that the loci of its vertices and of the intersection of its diagonals are all circles. CONSTRUCTIONS. Ex. 67. Show how to circumscribe about a given circle a triangle of given angles. (Suppose the figure drawn, and the points of contact joined to ⚫ the centre: what are the angles at the centre?) Ex. 68. Show how to inscribe in a given circle a triangle of given angles by considering what angles the sides subtend at the centre. Ex. 69. Show how to construct a triangle having given a side, the angle opposite to this side, and the radius of its inscribed circle. Is it always possible to construct a triangle with arbitrarily assigned values for these elements? [See Ex. 64.] Ex. 70. Show how to construct a quadrilateral ABCD, given AB, AC, AD, BAD, BCD. Ex. 71. Show how to construct a quadrilateral ABCD, given AB, BC, CA, AD, ▲ BDC. Ex. 72. Show how to construct a parallelogram, given the base and height and the angle (subtended by the base) at the intersection of the diagonals. COMMON TANGENTS. Ex. 73. Prove that the two exterior (or interior) common tangents of two circles are equal. Ex. 74. ST, S'T' are two exterior common tangents to two circles. Prove that SS' and TT' are parallel. Ex. 75. Express the length of a common tangent (exterior and interior) in terms of the radii of the circles and the distance between their centres. Ex. 76. The sum and difference of the radii of two circles which touch each other externally is given. Find by geometric construction the length of their common tangent. Ex. 77. The centres of two circles (which are outside each other) are fixed and the difference of their radii is constant. Show that the locus of the points of contact of exterior common tangents is two pairs of parallel lines. Ex. 78. The centres of two circles (which are outside each other) are fixed and the sum of the radii is constant. Prove that the interior common tangents constitute a system of equal parallel lines. Ex. 79. Show how to draw a line to intersect two given circles so that the parts intercepted by the circles may each be of given length. Ex. 80. Show how to draw a straight line through a point of intersection of two circles so that the part intercepted by the two circles may be (i) of given length, (ii) of maximum length. [Supposing it done, draw perpendiculars from the centres to the line and a parallel to the line from the centre of one of the circles.] ALTERNATE SEGMENT. TEx. 81. State and prove a form of converse to Theorem 51. Ex. 82. ACB is a segment of a circle containing an angle of half a right angle. Prove that the tangents at A and B are perpendicular. Ex. 83. A chord AB of a circle bisects the angle between the diameter through A, and the perpendicular from A upon the tangent at B. Ex. 84. AB, AC are two chords of a circle; BD is drawn parallel to the tangent at A, to meet AC in D; prove that LABD is equal or supplementary to BCD. Ex. 85. Two circles, whose centres are P, Q, intersect at A, B. The tangents at A to the two circles are drawn, and produced to cut the circumferences at D, C (see fig. 111). Prove that (i) the triangles BAC, BDA are equiangular, (ii) the angles APC, AQD are equal. Fig. 111. Ex. 86. A tangent is drawn at one end of an arc; and from the mid-point of the arc perpendiculars are drawn to the tangent, and the chord of the arc. Prove that they are equal. Ex. 87. ABC is a triangle inscribed in a circle and 'BT is drawn parallel to AC cutting the tangent at A in T; prove ▲ ABC= <BTA. Ex. 88. Two circles touch at P. Through P are drawn straight lines APB, CPD, cutting the one circle at A, C and the other at B, D. Prove that AC is parallel to BD. (Draw tangent at P.) Ex. 89. Two circles centres A and B intersect in C. Prove that the angles between the tangents to the two circles at C are equal or supplementary to ▲ ACB. Ex. 90. ABC is a triangle inscribed in a circle and the angle B is double the angle A. Prove that the bisector of the angle B is parallel to the tangent at C. Also, if this bisector meets AC in D, then BD=AD, Ex. 91. Two circles touch internally at A. Chords AP, AQ of the outer cut the inner in X and Y. Prove XY parallel to PQ. Ex. 92. Two circles touch internally at A. PQ a chord of the outer touches the inner at R. Prove AR bisects the angle PAQ. Ex. 93. With centre B, a point on the internal bisector of an angle ACD, a circle is drawn to pass through C and to cut CA in A. Prove that the circle circumscribing the triangle ABC will touch CD at C. Ex. 94. The diameter AB of a circle is produced to any point T; from T a line is drawn to touch the circle at P; the bisector of the angle ATP cuts BP, AP at R, Q respectively. Prove (i) PQ=PR; (ii) the angle TQP is half a right angle. Ex. 95. ABCDE is a regular pentagon inscribed in a circle. Prove that the tangent to the circle at A is parallel to CD. Ex. 96. D is a point on the side BC of a triangle ABC. A circle touches AB at B and passes through D, and a second circle touches AC at C and passes through D. Show that the locus of the other intersection of these circles is the circle circumscribed to the triangle ABC. Ex. 97. AB is any chord of a circle. If P and Q be the extremities of the diameter perpendicular to AB, show that AP, AQ bisect the two angles made by AB with the tangent at A. Ex. 98. The triangle ABC has the sides AB, AC equal, and D, E are the middle points of these sides. If the circle through D, E, A touches BC, prove that the triangle is right-angled. Ex. 99. Two circles intersect in A and B; PQ is a common tangent. Prove that the angles PAQ, PBQ are supplementary. Ex. 100. ABCD is a cyclic quadrilateral, whose diagonals intersect at E: a circle is drawn through A, B and E. Prove that the tangent to this circle at E is parallel to CD. Ex. 101. ABC is a triangle. DE parallel to BC cuts the sides in D and E. Prove that the circumcircles of the triangles ABC, ADE touch at A. Ex. 102. ABCD is a trapezium with AB || to CD and AC, BD meeting in O. Prove that the circles AOB, COD have external contact at O. Ex. 103. ABC is a triangle inscribed in a circle and the tangents at B and C meet in T. Prove that, if through T a straight line is drawn parallel to the tangent at A meeting AB, AC produced in F and G, then T is the mid-point of FG. SIMILAR FIGURES The fundamental theorem (Th. 52), like so many fundamental theorems, presents considerable difficulty. The proof given here, which assumes the ratios to be commensurable (see note on p. 112), will be found much simpler if the following preliminary work is discussed before considering the theorem. What is the definition of a fraction P where and q are integers (i.e. whole numbers)? It is that, if we divide the whole into q equal P parts and take p such parts, then the portion taken is of the whole. ト In fig. 112, suppose that AB is divided into q equal parts and that AH contains p such parts. A Then, according to the definition of a fraction, ૧ H B Fig. 112. See also that the above argument applies to fig. 113, in which H is a point in AB produced. (Note that will, in this case, be an improper fraction.). AH AB A B H Fig. 113. Conversely, for both figures, if AH P = equal parts, AH will contain p such parts. and if AB is divided into q When the above is clear, there should be no difficulty with the proof of Theorem 52. |