so that the vertices may meet in a point, and the area be everywhere doubled. 70. ABCD is a quadrilateral and ABCE is a parallelogram. Prove that A BCD=AACDEA CDE, according as A and E are on opposite sides or on the same side of CD. 71. ABCD, EBCF are two parallelograms on the same base and between the same parallels. If AC, BF meet in G, and BE, CD in H; show that GH (produced if necessary) passes through the midpoints of BC and DE. 72. ABCD, AHFE are two equal parallelograms having tho BAD identical with the < HAE; show that BE || CF || HD. 73. With the same data, if BC, EF (produced if necessary) meet in o, and DC, HF in P; show that the points A, O and P are in one straight line. 74. ABC is a triangle, P is any point in AB, and the PBCR is completed. If PR meet AC in Q, show that A APR=A ABQ. 75. ABC is a triangle, right-angled at A; squares ABFG, ACKH are described externally to the triangle on the sides AB, AC; BK, CF meet AC, AB at N, M; show that A AHN= A BCN, and A AGM = ДВСМ. 76. With the same data as in the previous Ex., show that ABP-ACP=CF2-BK?. 77. Show that the sum of the squares on the lines joining any point in its plane to the vertices of a rectangle is double the sum of the squares on the perpendiculars drawn from that point to the sides of the rectangle. 78. In the A ABC, D, E, F are points in BC, CA, AB such that AF+BD2+CE2=AE?+BF2+CD2; show that the perpendiculars drawn to the sides of the triangle through D, E, F are concurrent. (This is the converse to $ 20, Ex. 1.) 79. P, Q, R are the feet of the altitudes of the ABC, AP'I QR, BQ'IRP, CR'IPQ; show that AP', BQ', CR' are concurrent. (This may be shown to depend on the previous Deduction.) 80. If the perpendiculars from the vertices of the A ABC on the sides of a PQR are concurrent, show that the perpendiculars from the vertices of the A PQR on the sides of the A ABC are also concurrent. 81. D, E, F are the mid-points of the sides of the A ABC; OP, OQ, OR are three perpendiculars to the sides from a point O; P, Q', R' are points on BC, CA, AB such that DP=DP', etc.; show that the perpendiculars drawn to the sides of the A ABC through P,Q,R' are concurrent. 82. ABCD is a square whose diagonals intersect in O, and P is a point such that OP=OA; show that PA2+PB2+PC2+PD2=4 AB2. 83. P and Q are points on the same side of a straight line AB, R is the image of P in AB; (See $ 21, Lx. 1, Note); show that PQ<QR. 84. D, E, F are the images of A, B, C in a given straight line; show that A DEF=A ABC. 85. If from the extremities of the base of a triangle straight lines are drawn to any point in the bisector of the exterior vertical angle, their sum shall be greater than the sum of the two sides. 88. Of all triangles which have the same vertical angle and the same altitude, show that the isosceles has the smallest area and smallest base. 87. Of all rectilineal figures which have the same area and the same number of sides, show that the equilateral has the least perimeter. 88. Of all rectilineal figures which have the same perimeter and the same number of sides, show that the equilateral has the greatest area. 89. ABCDE is a pentagon in which AB=BC = CD = DE, and <B=2D = right angle. If o be the mid-point of AE, and BO, OD bejoined, prove that <BOD=right angle, and that the three parts into which the pentagon is divided can be put together so as to form a square. 90. On each side of a triangle equilateral triangles are described outwards ; show that their orthocentres are the vertices of an equilateral triangle. SS 22-30. EUCLID, I. Problems, 91. AX and AY are two straight lines, through A draw a straight line PQ such that _PAX= 2 QAY, 92. On a given straight line construct a regular hexagon. 93. Trisect an angle which is equal to (1) one-half, (2) one-fourth of a right angle. 94. Find the position of a point such that the sum of its distances from the vertices of a quadrilateral may be a minimum. 95. ABC is a triangle and D, E, F are the mid-points of its sides ; find the position of a point such that the sum of its distances from the six points A, B, C, D, E, F may be a minimum. 96. Through a given point A, draw three straight lines AP, AQ, AR such that they may be of given lengths, which satisfy the condition AP+AR>2AQ, and that Q may be the mid-point of PR. Loci. 97. ABCD is a parallelogram of constant area on a given base BC. Find the locus of the mid-point of AB. 98. Find the locus of the point of intersection of the diagonals of a parallelogram of given area described upon a given base. 99. ABC is a triangle and P is any point in BC; PQ, PR, are drawn parallel to AB, AC, meeting AC, AB in Q and R; AP, QR meet in 0; find the locus of O. 100. ABC is a triangle of constant area on a fixed base BC; its medians intersect in G; find the locus of G. 101. Find the locus of a point at which two equal segments of a given straight line subtend equal angles. 102. Find the locus of a point, the sum of whose distances from two parallel straight lines is equal to a given straight line greater than the distance between the parallel straight lines. 103. Find the locus of a point, the difference of whose distances from two parallel straight lines is equal to a given straight line less than the distance between the parallel straight lines. 104. Find the locus of a point, the sum of whose distances from two parallel straight lines is equal to the distance between the parallel straight lines. 105. Find the locus of a point, the difference of whose distances from two parallel straight lines is equal to the distance between the parallel straight lines. 106. Find the locus of a point, the sum of whose distances from two intersecting straight lines is equal to a given straight line. 107. Find the locus of a point, the difference of whose distances from two intersecting straight lines is equal to a given straight line. 108. A is a fixed point and XY a fixed straight line; P is any point in XY and PQ is drawn so that 2 APX = 2QPY, and so that AP+PQ = a given straight line; find the locus of Q. 109. The lengths of the sides of a parallelogram are given, and one side is fixed in position; find the locus of the point of intersection of the diagonals of the parallelogram. 110. ABC is a triangle ; PQ || BC meets AB, AC in P, Q; BQ, CP intersect in 0; find the locus of O. 111. The base BC of the A ABC is fixed, and the median from B is of constant length; find the locus of A. 112. Find the locus of the vertex of a right-angled triangle whose hypotenuse is a given straight line. 113. OX, OY are two straight lines given in position and at right angles to one another; A, B are points in Ox, OY, such that OA2+OB2 is constant; find the locus of the mid-point of AB. 114. On a fixed straight line AB, a A APB is described, rightangled at P; Q, R are the mid-points of AP, BP, and S is the midpoint of QR; find the locus of S. 115. ABCD is a parallelogram, and P is a point within the Z ABD; if the difference between the As PBC and PBD be constant, find the locus of P. 116. P is a point within a D ABCD, through P straight lines are drawn parallel to the sides of the parallelogram; if the difference between the parallelograms PA, PC remain constant, find the locus of P. N Analysis and Synthesis, etc. 117. A, B, C, D are fixed points; find a point equidistant from A and B, such that the sum of its distances from C and D shall be a minimum. 118. AX, AY are two straight lines; draw a straight line BC meeting AX, AY in B, C, so that BC shall be equal and parallel to a given straight line. 119. AX, AY are two straight lines; draw a straight line BC meeting AX, AY in B, C, so that BC shall be equal to a given straight line, and shall make equal angles with AX, AY. 120. A, B, C are three given points; through A draw a straight line PQ, such that the distances from A to the feet of the perpendiculars drawn to PQ from B and C may be equal. 121. Find a point P in the hypotenuse BC of the right-angled triangle ABC, such that if PQ be drawn perpendicular to AC meeting it in Q, BP=PQ. 122. AX, AY are two given straight lines and P is a given point. Through P draw a straight line meeting AX and AY in Q and R, so that (1) AQ=AR, 123. ABC is a triangle, find points P, Q in AB, AC, so that (1) PQ || BC and AP=CQ, 124. AB, AC are two given straight lines which form an acute angle, and P is a point within the angle. Find points Q and R in AB and AC, such that PQ, QR make equal angles with AB, and QR, RP make equal angles with AC. Show that your construction makes PQ+QR+RP a minimum. 125. E and F are two given points within the A ABD of the rectangle ABCD; ind four points P, Q, R, S in AB, BC, CD, DA, such that the rectilineal path EPQRSF may be a minimum. 126. E is a point within the A ABD of the rectangle ABCD; find four points P, Q, R, S in AB, BC, CD, DA, such that the rectilineal path EPQRSE may be a minimum, and show that the length of the path is independent of the position of E. |