63. Construct a rhombus having a given angle for one of its angles and having its sides each equal to a given straight line. 64. Construct a parallelogram equal to a given parallelogram in area and having its sides equal to two given straight lines. When is this impossible? 65. When two sides of a triangle are given the area is a maximum if they contain a right angle. 66. AB, AC are two given straight lines, and P is any point between them; show that the line through P, which forms with AB, AC the triangle of minimum area, is bisected at P. 67. In the figure of Prop. 1, if AB be produced both ways to meet the circumferences in M and N, the triangle MCN will be isosceles, and the angle MCN will be four times either of the angles CMN or CNM. 68. If one acute angle of a triangle be double or treble another, the triangle can be divided into two isosceles triangles. V.-On Loci. 69. Find the locus of a point equidistant from two given straight lines (i) when the lines meet, (ii) when the lines meet in an inaccessible point. 70. Find the locus of the middle point of a straight line drawn from a given point to meet a given straight line of unlimited length. 71. The locus of a point P such that the sum of the areas of two triangles PAB, PBC is constant, is a straight line parallel to AC. Distinguish the cases according as AB and BC are or are not in one straight line. 72. Find the locus of the point P which moves so that the sum of the areas of the two triangles PAB, PCD is constant. 73. Given the base and the difference of the squares on the sides of a triangle; find the locus of its vertex. 74. If the angular points of one parallelogram lie on the sides of another fixed parallelogram, find the locus of the intersection of the diagonals of the variable parallelogram. 75. Find the locus of a point the sum of whose distances from the four angular points of a convex quadrilateral is a minimum. VI.-On Parallel Lines and Projection. 76. If three parallel lines make equal intercepts on a fourth straight line which meets them, they will make equal intercepts on any other straight line which meets them. 77. The straight line AB is bisected at C, and AX, BY, CZ are drawn per pendicular to any other straight line PQ, which does not pass between A and B; prove that CZ is equal to half the sum of AX and BY What difference would it make if PQ did pass between A and B ? 78. If two triangles be on equal bases and between the same parallels, the intercepts made by the sides of the triangles on any straight line parallel to the base are equal. 79. The straight line, which joins the middle points of the oblique sides of a trapezium, is equal to half the sum of the parallel sides; and the part of this line intercepted between the diagonals of the trapezium is equal to half the difference of the parallel sides. 80. Through a given point draw a straight line so that the part intercepted between two given parallel straight lines may be equal to a given straight line. 81. If two straight lines are respectively parallel to two other straight lines, the angles made by the first pair are respectively equal to the angles made by the second pair. 82. Draw a straight line parallel to the base of a triangle so that the portion intercepted between the sides may be equal to a given straight line. 83. Straight lines that are equal and parallel have equal projections on any other straight line. State and prove also the converse of this. 84. Equal straight lines that have equal projections on another straight line are parallel to that line or make equal angles with it. VII.-On Parallelograms and Quadrilaterals. 85. If a straight line be drawn in any direction through one angle of a parallelogram, the perpendicular to it from the opposite angle of the parallelogram is equal to the sum or difference of the perpendiculars to it from the two remaining angles of the parallelogram. Distinguish the cases of the sum and difference of the perpendiculars. 86. Straight lines bisecting two opposite angles of a parallelogram are either coincident or parallel. 87. Straight lines bisecting two adjacent angles of a parallelogram intersect at right angles. 88. The diagonals of the rectangle formed by the bisectors of the angles of a parallelogram are parallel to the sides of the parallelogram and are equal to the difference between them. 89. If two parallelograms have two adjacent sides of the one equal to two adjacent sides of the other each to each, and have also the included angles equal, then the parallelograms shall be equal in all respects. 90. The straight lines which join the middle points of the sides of any quadrilateral form a parallelogram whose area is one half the area of the quadrilateral. 91. The quadrilaterals formed by either the four internal or the four external bisectors of the angles of any quadrilateral have their opposite angles together equal to two right angles. 92. If the opposite angles of a quadrilateral be equal, the opposite sides are equal. 93. The straight lines which join the middle points of the opposite sides of a quadrilateral, and the straight line which joins the middle points of the diagonals, are all concurrent and bisect each other. 94. Prove that the angle between the bisectors of two adjacent angles of a quadrilateral is half the sum of the two remaining angles of the quadrilateral. 95. A quadrilateral has two sides parallel and two sides equal but not parallel, show that the diagonals of the quadrilateral are equal. 96. If two opposite sides of a parallelogram be bisected and two straight lines be drawn from the points of bisection to two opposite angles, the two lines trisect the diagonal which passes through the other two angular points of the parallelogram. 97. If two quadrilaterals have three sides of the one equal to three sides of the other, each to each, and have also the angles contained by those sides equal, each to each, then the quadrilaterals shall be equal in all respects. VIII. On Areas and Squares. 98. Of all the parallelograms which can be formed with diameters of given lengths, the rhombus is the greatest. 99. Two rectangles are equal in area if two adjacent sides of the one are equal to two adjacent sides of the other. 100. The area of a triangle (or parallelogram) is equal to the sum or difference of the areas of two triangles (or parallelograms) on the same base (or equal bases) if the altitude of the former is equal to the sum or difference of the altitudes of the latter. show that the sum equal to the area of 101. ABCD is a parallelogram and P is any point; or difference of the areas of the triangles PAB, PAD is the triangle PAC. Distinguish the cases for the sum and difference. 102. ABC is a given triangle and P is a point in its base; if two lines be drawn through P parallel to the sides of the triangle, show that the parallelogram so formed is greatest when P is taken at the middle of the base. 103. The area of a quadrilateral is equal to the area of a triangle, two of whose sides are equal to the diagonals of the quadrilateral, the angle included by these sides being also equal to the angle between the diagonals. 104. If a square be inscribed in a triangle, twice the area of the triangle will be equal to the rectangle contained by a side of the square and a line equal to the sum of the base and altitude of the triangle. 105. Through one angle of a triangle draw a straight line to the opposite side cutting off from the triangle any given area. 106. Bisect a quadrilateral by a straight line drawn (i) through one of its angles, or (ii) through a given point in one of its sides. 107. Trisect a parallelogram by straight lines drawn (i) through one of its angles, or (ii) through a given point in one of its sides. 108. If a straight line be divided into any two parts, the square on the whole line is greater than the sum of the squares on the two parts. 109. Divide a given straight line into two parts so that the square on one part may be double the square on the other part. 110. Divide a straight line into two parts, the difference of whose squares shall be equal to a given square. 111. In a right-angled triangle the equilateral triangle described on the hypotenuse is equal to the sum of the equilateral triangles described on the two sides. 112. If any parallelograms ABDE, ACFG be described externally on the sides AB, AC of any triangle ABC, and if DE, FG, or these lines produced, meet in H; then the parallelogram BKLC, described on BC and having its sides BK, CL equal and parallel to HA, will be equal to the sum of the parallelograms AD and AF. PRACTICAL EXERCISES ON BOOK I. Note on Euclid's Constructions. In practical work Propositions 9, 10, 11, 22, 31 of Book I. should be replaced by the methods given in the Preliminary Course in Problems V., IV., II., XIV. respectively. Prop. 23 is usually modified as follows:-With centre D describe an arc cutting DC in H and DE in K; with centre A and the same radius describe an arc cutting AB in L; from the latter arc cut off a portion LM equal to the arc HK, measuring by the compasses; join AM. 1. Draw the figure of I. 15, making ▲ DEB = 40°. Measure LB DEA, AEC, CEB. Bisect each of the four angles at E, and verify that the bisectors form two perpendicular straight lines. [Cf. Props. 13, 15, and Ex. 1, p. 40.] 2. Draw a line AB of length 1.5". Draw the locus of all points distant 1" from A; also the locus of all points distant 1.5′′ from B. Find two points, each of which lies 1" from A and 1.5" from B. 3. Draw a line AB of length 30 mm. Find two points, each of which lies 20 mm. from A and from B; also two points, each of which lies 35 mm. from A and from B. 4. Draw a line AB of length 30 mm. are equidistant from A and B. Draw the locus of all points which = 3 cm., BC = 4 cm., CA 4 cm. 5. Construct a AABC such that AB Draw the locus of all points which are equidistant from A and B ; also the locus of all points which are equidistant from B and C. Find a point which is equidistant from A, B, and C, and measure its distance from any one of these points. 6. Construct a A ABC such that AB = 2", BC = = 2.5", CA 1.5". Find a point on AB which is equidistant from B and C, and measure its distance from either. EUC. 113 I |