the centre of the third circle from the centres of the other two is invariable. 10. Draw two concentric circles, such that those chords of the outer circle, which touch the inner, may equal its diameter. 11. If the sides of a quadrilateral inscribed in a circle be bisected and the middle points of adjacent sides joined, the circles described about the triangles thus formed are all equal and all touch the original circle. 12. Draw a tangent to a circle which shall be parallel to a given finite straight line. 13. Describe a circle, which shall have a given radius, and its centre in a given straight line, and shall also touch another straight line, inclined at a given angle to the former. 14. Find a point in the diameter produced of a given circle, from which, if a tangent be drawn to the circle, it shall be equal to a given straight line. 15. Two equal circles intersect in the points A, B, and through B a straight line CBM is drawn cutting them again in C, M. Shew that if with centre C and radius BM a circle be described, it will cut the circle ABC in a point L such that arc AL-arc AB. Shew also that LB is the tangent at B. 16. AB is any chord and AC a tangent to a circle at A ; CDE a line cutting the circle in D and E and parallel to AB. Shew that the triangle ACD is equiangular to the triangle EAB. 17. Two equal circles cut one another in the points A, B ; BC is a chord equal to AB; shew that AC is a tangent to the other circle. 18. In any two circles, which cut one another, the straight line joining the extremities of any two parallel radii cuts the line joining the centres in the same point. 19. A, B are two points; with centre B describe a circle, such that its tangent from A shall be equal to a given line, 20. If perpendiculars be dropped from the angular points of a triangle on the opposite sides, shew that the sum of the squares on the sides of the triangle is equal to twice the sum of the rectangles, contained by the perpendiculars and that part of each intercepted between the angles of the triangles and the point of intersection of the perpendiculars. 21. When two circles intersect, their common chord bisects their common tangent. 22. Two circles intersect in A and B. Two points C and D are taken on one of the circles; CA, CB meet the other circle in E, F, and DA, DB meet it in G, H: shew that FG is parallel to EH, and FH to EG. 23. A and B are fixed points, and two circles are described passing through them; CP, CP' are drawn from a point C on AB produced, to touch the circles in P, P'; shew that CP=CP. 24. From each angular point of a triangle a perpendicular is let fall upon the opposite side; prove that the rectangles contained by the segments, into which each perpendicular is divided by the point of intersection of the three, are equal to each other. 25. If from a point without a circle two equal straight lines be drawn to the circumference and produced, shew that they will be at the same distance from the centre. 26. Let O, O' be the centres of two circles which cut each other in A, A'. Let B, B' be two points, taken one on each circumference. Let C, C' be the centres of the circles BAB', BA'B'. Then prove that the angle CBC is equal to the angle OA'O'. 27. The common chord of two circles is produced to any point P; PA touches one of the circles in A; PBC is any chord of the other: shew that the circle which passes through A, B, C touches the circle to which PA is a tangent. 28. Given the base of a triangle, the vertical angle, and the length of the line drawn from the vertex to the middle point of the base construct the triangle, 29. If a circle be described about the triangle ABC, and a straight line be drawn bisecting the angle BAC and cutting the circle in D, shew that the angle DCB will be equal to half the angle BAC. 30. If the line AD bisect the angle A in the triangle ABC, and BD be drawn without the triangle making an angle with BC equal to half the angle BAC, shew that a circle may be described about ABCD. 31. Two equal circles intersect in A, B: PQT perpendicular to AB meets it in Tand the circles in P, Q. AP, BQ meet in R; AQ, BP in S; prove that the angle RTS is bisected by TP. 32. If the angle, contained by any side of a quadrilateral and the adjacent side produced, be equal to the opposite angle of the quadrilateral, prove that any side of the quadrilateral will subtend equal angles at the opposite angles of the quadrilateral. 33. If DE be drawn parallel to the base BC of a triangle ABC, prove that the circles described about the triangles ABC and ADE have a common tangent at A. 34. Describe a square equal to the difference of two given squares. 35. If tangents be drawn to a circle from any point without it, and a third line be drawn between the point and the centre of the circle, touching the circle, the perimeter of the triangle formed by the three tangents will be the same for all positions of the third point of contact. 36. If on the sides of any triangle as chords, circles be described, of which the segments external to the triangle contain angles respectively equal to the angles of a given triangle, those circles will intersect in a point. 37. Prove that if ABC be a triangle inscribed in a circle, such that BA=BC, and AA′ be drawn parallel to BC, meeting the circle again in A', and A'B be joined cutting AC in E, BA touches the circle described about the triangle AEA'. 38. Describe a circle, cutting the sides of a given square, so that its circumference may be divided at the points of intersection into eight equal arcs. 39. A is the extremity of the diameter of a circle, O any point in the diameter. The chord which is bisected at O subtends a greater or less angle at A than any other chord through O, according as O and A are on the same or opposite sides of the centre. 40. Shew that the square on the tangent drawn from any point in the outer of two concentric circles to the inner equals the difference of the squares on the tangents, drawn from any point, without both circles, to the circles. 41. If from a point without a circle, two tangents PT, PT, at right angles to one another, be drawn to touch the circle, and if from T any chord TQ be drawn, and from T' a perpendicular TM be dropped on TQ, then T'M=QM. 42. Find the loci : (1.) Of the centres of circles passing through two given points. (2.) Of the middle points of a system of parallel chords in a circle. (3.) Of points such that the difference of the distances of each from two given straight lines is equal to a given straight line. (4.) Of the centres of circles touching a given line in a given point. (5.) Of the middle points of chords in a circle that pass through a given point. (6.) Of the centres of circles of given radius which touch a given circle. (7.) Of the middle points of chords of equal length in a circle. (8.) Of the middle points of the straight lines drawn from a given point to meet the circumference of a given circle. 43. If the base and vertical angle of a triangle be given, find the locus of the vertex. 44. A straight line remains parallel to itself while one of its extremities describes a circle. What is the locus of the other extremity? 45. A ladder slips down between a vertical wall and a horizontal plane : what is the locus of its middle point? 46. AB is the diameter of a circle; ACD is a chord produced to D, so that AC-CD. Find the locus of the point in which BC and the line joining D to the centre intersect. 47. ABC is a line drawn from a point A, without a circle, to meet the circumference in B and C. Tangents are drawn to the circle at B and C which meet in D. What is the locus of D? 48. Two circles intersect in the points A, B; any straight line CDEF is drawn cutting the circles in C, D, E, F; prove that AC intersects BD and AE intersects BF in points, which lie on a circle passing through A and B. 49. The angular points A, C of a parallelogram ABCD move on two fixed straight lines OA, OC, whose inclination is equal to the angle BCD; shew that the points B, D will move on two fixed straight lines passing through O. 50. On the line AB is described the segment of a circle, in the circumference of which any point C is taken. If AC, BC be joined, and a point P taken in AC so that CP is equal to CB, find the locus of P. 51. Find the locus of the centre of the circles circumscribing two trapeziums, into which a parallelogram is divided by any line equal to one of its shorter sides. 52. If a parallelogram be described having the diameter of a given circle for one of its sides, and the intersection of its diagonals on the circumference, shew that the extremity of each of the diagonals moves on the circumference of another circle of double the diameter of the first. 53. One diagonal of a quadrilateral inscribed in a circle is fixed, and the other of constant length. Shew that the sides will meet, if produced, on the circumference of a fixed circle. |