74. Given one of the angles at the base of a triangle, the base itself, and the sum of the two remaining sides, to construct the triangle. 75. Given the base, an angle adjacent to the base, and the difference of the sides of a triangle, to construct it. 76. Given one angle, a side opposite to it, and the difference of the other two sides; to construct the triangle. 77. Given the base and the sum of the two other sides of a triangle, construct it so that the line which bisects the vertical angle shall be parallel to a given line. V. 78. From a given point without a given straight line, to draw a line making an angle with the given line equal to a given rectilineal angle. 79. Through a given point A, draw a straight line ABC meeting two given parallel straight lines in B and C, such that BC may be equal to a given straight line. 80. If the line joining two parallel lines be bisected, all the lines drawn through the point of bisection and terminated by the parallel lines are also bisected in that point. 81. Three given straight lines issue from a point: draw another straight line cutting them so that the two segments of it intercepted between them may be equal to one another. 82. AB, AC are two straight lines, B and C given points in the same; BD is drawn perpendicular to AC, and DE perpendicular to AB; in like manner CF is drawn perpendicular to AB, and FG to AC. Shew that EG is parallel to BC. 83. ABC is a right-angled triangle, and the sides AC, AB are produced to D and F; bisect FBC and BCD by the lines BE, CE, and from E let fall the perpendiculars EF, ED. Prove (without assuming any properties of parallels) that ADEF is a square. 84. Two pairs of equal straight lines being given, shew how to construct with them the greatest parallelogram. 85. With two given lines as diagonals describe a parallelogram which shall have an angle equal to a given angle. Within what limits must the given angle lie? 86. Having given one of the diagonals of a parallelogram, the sum of the two adjacent sides and the angle between them, construct the parallelogram. 87. One of the diagonals of a parallelogram being given, and the angle which it makes with one of the sides, complete the parallelogram, so that the other diagonal may be parallel to a given line. 88. ABCD, ABCD are two parallelograms whose corresponding sides are equal, but the angle 4 is greater than the angle A', prove that the diameter AC is less than A'C', but BD greater than B'D'. 89. If in the diagonal of a parallelogram any two points equidistant from its extremities be joined with the opposite angles, a figure will be formed which is also a parallelogram. 90. From each angle of a parallelogram a line is drawn making the same angle towards the same parts with an adjacent side, taken always in the same order; shew that these lines form another parallelogram similar to the original one. = 91. Along the sides of a parallelogram taken in order, measure AA' = BB' = CC DD': the figure A'B'C'D' will be a parallelogram. 92. On the sides AB, BC, CD, DA, of a parallelogram, set off AE, BF, CG, DH, equal to each other, and join AF, BG, CH,DE: these lines form a parallelogram, and the difference of the angles AFB, BGC, equals the difference of any two proximate angles of the two parallelograms. 93. OB, OC are two straight lines at right angles to each other, through any point P any two straight lines are drawn intersecting OB, OC, in B, B, C, C', respectively. If D and D' be the middle points of BB and CC, shew that the angle B'PD' is equal to the angle DOD. 94. ABCD is a parallelogram of which the angle C' is opposite to the angle A. If through A any straight line be drawn, then the distance of Cis equal to the sum or difference of the distances of B and of D from that straight line, according as it lies without or within the parallelogram. 95. Upon stretching two chains AC, BD, across a field ABCD, I find that BD and AC make equal angles with DC, and that AC makes the same angle with AD that BD does with BC; hence prove that AB is parallel to CD. 96. To find a point in the side or side produced of any parallelogram, such that the angle it makes with the line joining the point and one extremity of the opposite side, may be bisected by the line joining it with the other extremity. 97. When the corner of the leaf of a book is turned down a second time, so that the lines of folding are parallel and equidistant, the space in the second fold is equal to three times that in the first. VI. 98. If the points of bisection of the sides of a triangle be joined. the triangle so formed shall be one-fourth of the given triangle. 99. If in the triangle ABC, BC be bisected in D, AD joined and bisected in E, BE joined and bisected in F, and CF joined and bisected in G; then the triangle EFG will be equal to one-eighth of the triangle ABC. 100. Shew that the areas of the two equilateral triangles in Prob. 59, p. 78, are respectively, one-third and one-seventh of the area of the original triangle. 101. To describe a triangle equal to a given triangle, (1) when the base, (2) when the altitude of the required triangle is given. 102. To describe a triangle equal to the sum or difference of two given triangles. 103. Upon a given base describe an isosceles triangle equal to a given triangle. 104. Describe a right-angled triangle equal to a given triangle ABC. 105. To a given straight line apply a triangle which shall be equal to a given parallelogram and have one of its angles equal to a given rectilineal angle. 106. Transform a given rectilineal figure into a triangle whose vertex shall be in a given angle of the figure, and whose base shall be in one of the sides. 107. Divide a triangle by two straight lines into three parts which when properly arranged shall form a parallelogram whose angles are of a given magnitude. 108. Shew that a scalene triangle cannot be divided into two parts which will coincide. 109. If two sides of a triangle be given, the triangle will be greatest when they contain a right angle. 110. Of all triangles having the same vertical angle, and whose bases pass through a given point, the least is that whose base is bisected in the given point. 111. Of all triangles having the same base and the same perimeter, that is the greatest which has the two undetermined sides equal. 112. Divide a triangle into three equal parts, (1) by lines drawn from a point in one of the sides: (2) by lines drawn from the angles to a point within the triangle: (3) by lines drawn from a given point within the triangle. In how many ways can the third case be done? 113. Divide an equilateral triangle into nine equal parts. 114. Bisect a parallelogram, (1) by a line drawn from a point in one of its sides: (2) by a line drawn from a given point within or without it: (3) by a line perpendicular to one of the sides: (4) by a line drawn parallel to a given line. 115. From a given point in one side produced of a parallelogram, draw a straight line which shall divide the parallelogram into two equal parts. 116. To trisect a parallelogram by lines drawn (1) from a given point in one of its sides, (2) from one of its angular points. VII. 117. To describe a rhombus which shall be equal to any given quadrilateral figure. 118. Describe a parallelogram which shall be equal in area and perimeter to a given triangle. 119. Find a point in the diagonal of a square produced, from which if a straight line be drawn parallel to any side of the square, and meeting another side produced, it will form together with the produced diagonal and produced side, a triangle equal to the square. 120. If from any point within a parallelogram, straight lines be drawn to the angles, the parallelogram shall be divided into four triangles, of which each two opposite are together equal to one-half of the parallelogram. 121. If ABCD be a parallelogram, and E any point in the diagonal AC, or AC produced; shew that the triangles EBC, EDC, are equal, as also the triangles EBA and EBD. 122. ABCD is a parallelogram, draw DFG meeting BC in F and AB produced in G; join AF, CG; then will the triangles A.B.F, CFG be equal to one another. 123. ABCD is a parallelogram, E the point of intersection of its diagonals, and K any point in AD. If KB, KC be joined, shew that the figure BKEC is one-fourth of the parallelogram. 124. Let ABCD be a parallelogram, and O any point within it, through O draw lines parallel to the sides of ABCD, and join O, OC; prove that the difference of the parallelograms DO, BŎ is twice the triangle OAC. 125. The diagonals AC, BD of a parallelogram intersect in O and P is a point within the triangle AOB; prove that the difference of the triangles APB, CPD is equal to the sum of the triangles APC, BPD. 126. If K be the common angular point of the parallelograms about the diameter AC (fig. Euc. 1. 43.) and BD be the other diameter, the difference of these parallelograms is equal to twice the triangle BKD. 127. The perimeter of a square is less than that of any other parallelogram of equal area. 128. Shew that of all equiangular parallelograms of equal perimeters, that which is equilateral is the greatest. 129. Prove that the perimeter of an isosceles triangle is greater than that of an equal right-angled parallelogram of the same altitude. VIII. 130. If a quadrilateral figure is bisected by one diagonal, the second diagonal is bisected by the first. 131. If two opposite angles of a quadrilateral figure are equal, shew that the angles between opposite sides produced are equal. 132. Prove that the sides of any four-sided rectilinear figure are together greater than the two diagonals. 133. The sum of the diagonals of a trapezium is less than the sum of any four lines which can be drawn to the four angles, from any point within the figure, except their intersection. 134. The longest side of a given quadrilateral is opposite to the shortest; shew that the angles adjacent to the shortest side are together greater than the sum of the angles adjacent to the longest side. 135. Give any two points in the opposite sides of a trapezium, inscribe in it a parallelogram having two of its angles at these points. 136. Shew that in every quadrilateral plane figure, two parallelograms can be described upon two opposite sides as diagonals, such that the other two diagonals shall be in the same straight line and equal. 137. Describe a quadrilateral figure whose sides shall be equal to four given straight lines. What limitation is necessary? 138. If the sides of a quadrilateral figure be bisected and the points of bisection joined, the included figure is a parallelogram, and equal in area to half the original figure. 139. A trapezium is such, that the perpendiculars let fall on a diagonal from the opposite angles are equal. Divide the trapezium into four equal triangles, by straight lines drawn to the angles from a point within it. 140. If two opposite sides of a trapezium be parallel to one another, the straight line joining their bisections, bisects the trapezium. 141. If of the four triangles into which the diagonals divide a trapezium, any two opposite ones are equal, the trapezium has two of its opposite sides parallel. 142. If two sides of a quadrilateral are parallel but not equal, and the other two sides are equal but not parallel, the opposite angles of the quadrilateral are together equal to two right angles: and conversely. 143. If two sides of a quadrilateral be parallel, and the line joining the middle points of the diagonals be produced to meet the other sides; the line so produced will be equal to half the sum of the parallel sides, and the line between the points of bisection equal to half their difference. 144. To bisect a trapezium, (1) by a line drawn from one of its angular points: (2) by a line drawn from a given point in one side. 145. To divide a square into four equal portions by lines drawn from any point in one of its sides. 146. It is impossible to divide a quadrilateral figure (except it be a parallelogram) into equal triangles by lines drawn from a point within it to its four corners. IX. 147. If the greater of the acute angles of a right-angled triangle, be double the other, the square on the greater side is three times the square on the other. 148. Upon a given straight line construct a right-angled triangle such that the square on the other side may be equal to seven times the square on the given line. 149. If from the vertex of a plane triangle, a perpendicular fall upon the base or the base produced, the difference of the squares on the sides is equal to the difference of the squares on the segments of the base. 150. If from the middle point of one of the sides of a right-angled triangle, a perpendicular be drawn to the hypotenuse, the difference of the squares on the segments into which it is divided, is equal to the square on the other side. that 151. If a straight line be drawn from one of the acute angles of a right-angled triangle, bisecting the opposite side, the square upon line is less than the square upon the hypotenuse by three times the square upon half the line bisected. 152. If the sum of the squares on the three sides of a triangle be equal to eight times the square on the line drawn from the vertex to the point of bisection of the base, then the vertical angle is a right angle. 153. If a line be drawn parallel to the hypotenuse of a rightangled triangle, and each of the acute angles be joined with the points where this line intersects the sides respectively opposite to them, the squares on the joining lines are together equal to the squares on the hypotenuse and on the line drawn parallel to it. |