PROBLEMS. BOOK III. 1. DESCRIBE a circle of given radius, which shall pass through two given points. 2. If with the vertex of an isosceles triangle, as centre, a circle be described cutting the base or base produced, the parts of it intercepted between the circle and the extremities of the base will be equal. 3. If two circles cut each other, any two parallel lines drawn through the points of section to cut the circles are equal. 4. If two circles cut each other, draw through one of the points of section a line which shall be terminated in the circumferences and be bisected in that point. 5. A chord PAQ cuts the diameter of a circle in A in an angle which is half a right angle: shew that the squares of AP and AQ are together double of the square of the radius. 6. If two chords intersect in a circle, the difference of their squares is equal to the difference of the squares of the difference of the segments. 7. Two parallel chords in a circle are respectively six and eight inches in length, and are one inch apart: how many inches in length is the diameter ? 8. Draw a line cutting two concentric circles, so that the part of it intercepted by the circumference of the greater may be double the part intercepted by that of the less. 9. If two circles cut each other, the greatest line that can be drawn through the point of intersection is that which is parallel to the line joining their centres. 10. Describe three equal circles touching one another, and also another which shall touch them all three. 11. How many equal circles can be described around another circle of the same magnitude, touching it and one another? 12. Describe a circle which shall pass through a given point, and touch a given circle in a given point, the two points not being in a tangent to the given circle. 13. Describe a circle which shall touch a given circle in a given point, and also touch a given straight line. 14. If from any point without a circle two lines be drawn making equal angles with the line through the centre from that point, they will cut off equal segments from the circle. 15. Through a given point within a circle draw the least possible chord. 16. Of all lines which touch the interior and are bounded by the exterior of two circles which touch internally, the greatest is that which is parallel to the common tangent. 17. Shew that the two tangents to a circle drawn from the same point without it are equal to one another; and hence prove that the sums of the opposite sides of any quadrilateral described about a circle are equal, and the angles subtended at the centre of the circle by any two opposite sides together equal to two right angles. 18. If any line be drawn touching a circle, the part of it intercepted between the tangents at the extremities of any diameter subtends at the centre a right angle. 19. In the diameter of a circle produced determine a point from which a tangent drawn to the circle shall be equal to the diameter. 20. Describe a circle which shall pass through a given point, have a given radius, and touch a given line. 21. Describe a circle whose centre shall be in the perpendicular of a given right-angled triangle, and which shall pass through the right angle and touch the hypo thenuse. 22. A is any point in the diameter (or diameter produced) of a circle, whose centre is O, OB a radius perpendicular to the diameter: if AB cut the circle in P, and the tangent at P cut AO in C, shew that AC=CP. 23. A common tangent is drawn to two circles which touch externally if a circle be described on that part of it which lies between the points of contact, as diameter, it will pass through the point of contact of the two circles, and be touched by the line joining their centres. 24. Describe a circle with given radius and its centre in a given line, which shall touch another given line. 25. Describe a circle, which shall touch a given line in a given point, and also touch a given circle. 26. Draw a common tangent to two circles, when the points of contact are (i) on the same side, and (ii) on opposite sides, of the line joining their centres. 27. Draw a line which shall touch a given circle, and make with a given line a given angle. 28. Describe two circles of given radii, which shall touch each other, and the same given line on the same side of it. 29. If two circles touch each other, and parallel diameters be drawn, then lines which join the extremities of these diameters will pass through the point of contact. 30. The line, drawn from the vertex of an equilateral triangle to meet the circumscribing circle in any point, is equal to the sum or difference of the two lines drawn from the extremities of the base to that point, according as it does or does not cut the base. 31. AB, AC are any two chords of a circle, D, E, the bisections of the arcs AB, AC: let DE cut AB, AC in F, G, and shew that AF-AG. 32. If the opposite angles of a quadrilateral be together equal to two right angles, shew that a circle may be described about it, and find its centre and radius. 33. ABCD is a parallelogram; draw CE perpendicular to the diagonal BD, and shew that perpendiculars upon AB, AD at the points B, D, will intersect in CE. 34. The circles described on the three sides of a triangle, so as to pass through the points of intersection of the perpendiculars upon them from the opposite angles, are equal to each other. 35. Two circles intersect in A, B, the centre of one being in the circumference of the other: draw any chord ACD cutting them both, and shew that CB= CD. 36. If from any two points in the circumference of a circle there be drawn two lines to a point in any tangent to the circle, they will make the greatest angle when drawn to the point of contact. 37. Given three points in a circle: shew how we may find any number of other points, without knowing the position of the centre. 38. If through the angles of a quadrilateral, lines bisecting them be drawn, the points in which each line intersects the adjacent ones will all lie in the circumference of a circle. 39. ABC is a semicircle, ADC a quadrant, upon the same line AC and on the same side of it; from any point B in the semicircle draw BA, BDC, and shew that BA and BD are equal, and that the longer only of the lines AB, AC, can cut the circle ADC. 40. If any chord of a circle be bisected by another and produced to meet the tangents at the extremities of the bisecting line, the parts intercepted between the tangents and the circumferences are equal. M 41. From the extremities A, C of a given circular arc, equal arcs AB, CD are measured in opposite directions: shew that the chords AC, BD are parallel. 42. The arcs intercepted between any two parallel chords of a circle are equal; and if any two chords of a circle intersect each other, the sum of the arcs intercepted by them is equal to the sum of the arcs intercepted by diameters parallel to them. 43. A, B, C, A', B', C' are points in the circumference of a circle: if AB, AC be respectively parallel to A'B', A'C', shew that BC' is parallel to B'C. 44. If two equal circles cut each other, and from either point of intersection a circle be described cutting them, the point where this circle cuts them and the other point of intersection of the equal circles are in the same straight line. 45. If two equal circles cut each other and from either point of intersection a line be drawn cutting the circumferences, the part of it between them will be bisected by the circle, whose diameter is the common chord of the equal circles. 46. If two circles cut each other, and any two points be taken in the circumference of one through which lines are drawn from the points of intersection cutting that of the other, the lines, joining the points of section with the latter circle of those drawn through the same point, will be equal. 47. A, B are given points: if AC, BC are drawn, making a given angle with each other, shew that the line bisecting that angle passes through a fixed point. 48. If perpendiculars be dropped from the extremities of any diameter upon any chord of a circle, the parts of the chord, intercepted between them and the circle, will be equal, and the less perpendicular shall be equal to the segment of the greater contained between the chord and circumference. |