ERRATA. Page 73, question 415 is wrong. 107, line 6 from the bottom, for "all values of @" read values of @ 108, line 10, for +336n+162, read 336n+164. 109, line 5, dele the second "that." 211, question 1267, insert "Prove that." 273, line 9, for x+q1, x+a, read x+b1, x+b.. 289, question 1686, for u1-2" read un-1; and for 1-2, read -1. line 3 from the bottom, for +... read GEOMETRY. 1. A point O is taken within a polygon ABC...KL; prove that OA, OB,...OL are together greater than half the perimeter of the polygon. 2. Two triangles are on the same base and between the same parallels; through the point of intersection of their sides is drawn a straight line parallel to the base and terminated by the sides which do not intersect: prove that the segments of this straight line are equal. 3. The sides AB, AC of a triangle are bisected in D, E, and CD, BE intersect in F: prove that the triangle BFC is equal in area to the quadrangle ADFE. 4. AB, CD are two parallel straight lines, E the middle point of CD, and F, G the respective points of intersection of AC, BE, and of AE, BD: prove that FG is parallel to AB. 5. Through the angular points of a triangle are drawn three parallel straight lines terminated by the opposite sides': prove that the triangle formed by joining the ends of these lines will be double of the original triangle. 6. If a, b, c be the middle points of the sides of a triangle ABC, and if through A, B, C be drawn three parallels to meet bc, ca, ab respectively in A', B', C', the sides of the triangle A'B'C' will pass through A, B, C respectively, and the triangle ABC will be double of the triangle A'B'C'. 7. In a right-angled triangle the straight line joining the right angle to the centre of the square on the hypotenuse will bisect the right angle. 8. Through the vertex of an equilateral triangle is drawn a straight line terminated by the two straight lines drawn through the ends of the base at right angles to the base, and on this straight line as base is described another equilateral triangle: prove that the vertex will lie either on the base of the former or on a fixed straight line parallel to that base. 1 All straight lines are supposed to be produced if necessary. W. P. 1 1 2 9. Through the angle C of a parallelogram ABCD is drawn a straight line meeting the two sides AB, AD in P, Q: prove that the rectangle under BP, DQ is of constant area. 10. In any quadrangle the squares on the sides together exceed the squares on the diagonals by the square on twice the line joining the middle points of the diagonals. 11. If a straight line be divided in extreme and mean ratio and produced so that the part produced is equal to the smaller of the segments, the rectangle contained by the whole line thus produced, and the part produced together with the square on the given line will be equal to four times the square on the larger segment. 12. Two equal circles touch at A, a circle of double the radius is drawn having internal contact with one of them at B and cutting the other in two points: prove that the straight line AB will pass through one of the points of section. 13. Two straight lines inclined at a given angle are drawn touching respectively two given concentric circles: their point of intersection will lie on one of two fixed circles concentric with the given circles. 14. A chord CD is drawn at right angles to a fixed diameter AB of a given circle, and DP is any other chord meeting AB in Q: prove that the angle PCQ is bisected by either CA or CB. 15. AB is the diameter of a circle, P a point on the circle, PM perpendicular on AB; on AM, MB as diameters are described two circles meeting AP, BP in Q, R respectively: prove that QR will touch both circles. 16. Given two straight lines in position and a point equidistant from them, prove that any circle through the given point and the point of intersection of the two given lines will intercept on the lines segments whose sum or whose difference will be equal to a given length. 17. A triangle circumscribes a circle and from each point of contact is drawn a perpendicular to the straight line joining the other two: prove that the straight lines joining the feet of these perpendiculars will be parallel to the sides of the original triangle. 18. From a fixed point 0 of a given circle are drawn two chords OP, OQ equally inclined to a fixed chord: prove that PQ will be fixed in direction. 19. Through the ends of a fixed chord of a given circle are drawn two other chords parallel to each other: prove that the straight line joining the other ends of these chords will touch a fixed circle. 20. Two circles with centres A, B cut each other at right angles and their common chord meets AB in C; DE is a chord of the first circle passing through B: prove that A, D, E, C lie on a circle. 21. Four fixed points lie on a circle, and two other circles are drawn touching each other, one passing through two fixed points of the four and the other through the other two: prove that their point of contact lies on a fixed circle. 22. A circle A passes through the centre of a circle B: prove that their common tangents will touch A in points lying on a tangent to B; and conversely. 23. On the same side of a straight line AB are described two segments of circles, AP, AQ are chords of the two segments including an angle equal to that between the tangents to the two circles at 4: prove that P, Q, B are in one straight line. 24. The centre A of a circle lies on another circle which cuts the former in B, C; AD is a chord of the latter circle meeting BC in E and from D are drawn DF, DG to touch the former circle: prove that G, E, F lie on one straight line. 25. If the opposite sides of a quadrangle inscribed in a circle be produced to meet in P, Q, and if about two of the triangles so formed circles be described meeting again in R: P, R, Q will be in one straight line. 26. Two circles intersect in A and through A any two straight lines BAC, B'AC' are drawn terminated by the circles: prove that the chords BB, CC' of the two circles are inclined at a constant angle. 27. If two circles touch at A and PQ be any chord of one circle touching the other, the sum or the difference of the chords AP, AQ will bear to the chord PQ a constant ratio. 28. Four points A, B, C, P are taken on a circle and chords PA', PB', PC' drawn parallel respectively to BC, CA, AB: prove that the angles APA', BPB', CPC have common internal and external bisectors. 29. Two circles are drawn such that their two common points and the centre of either are corners of an equilateral triangle, P is one common point and PQ, PQ' tangents at P terminated each by the other circle: prove that QQ' will be a common tangent. 30. On a fixed diameter AB of a given circle is taken a fixed point C from which perpendiculars are let fall on the straight lines joining A, B to any point of the circle: prove that the straight line joining the feet of these perpendiculars will pass through a fixed point. [If D be this fixed point and O the centre, the rectangle under OC, OD will be half the sum of the squares on OC, OA.] 31. Four points are taken on a circle and the three pairs of straight lines which can be drawn through the four points intersect respectively in E, F, G: prove that the three pairs of straight lines which bisect the angles at E, F, G respectively will be in the same directions. 32. Through one point of intersection of two circles is drawn a straight line at right angles to their common chord and terminated by the circles, and through the other point is drawn a straight line equally inclined to the straight lines joining that point to the extremities of the former straight line: prove that the tangents to the two circles at the points on this latter straight line will intersect in a point on the common chord. |