BOOK III. 2. Through a given point within a circle draw the least possible chord. 3. Through a given point within a circle draw a chord which shall be bisected in that point. 4. Bisect a circle by a straight line drawn through a given point without it. 5. If two chords be drawn in a circle through any point in a diameter equally inclined to it, they shall be equal to one another. 6. A chord PQ of a circle is cut by any diameter AB; prove that the difference between the areas of the triangles PAQ, PBQ is constant. 7. If from a point within a circle two straight lines are drawn to the circumference equal to one another, the centre lies in the straight line bisecting the angle between them. 8. If from a point without a circle two straight lines are drawn to the circumference equal to one another, the centre lies in the straight line bisecting the angle between thein. 9. Describe a circle passing through two given points and bisecting the circumference of a given circle. 10. Draw a tangent to a circle parallel to a given straight line. II. Draw a straight line touching a given circle and making a given angle with a given straight line. 12. ABC is a triangle, having an acute angle at A; shew that the square on BC is < the squares on BA, AC by twice the square on a line drawn from A to touch the circle described on BC as diameter. 13. Find a point such that the tangents drawn from it to two given circles may contain a given angle. 14. Find a point from which if straight lines be drawn to touch three given circles, neither of which lies within the other, the tangents so drawn shall be equal. 15. Find the locus of points from which the tangents drawn to two given circles are equal. . 16. Describe a circle passing through one of the angular points of a square, and touching the sides containing the opposite angle. 17. Through a given point draw a straight line cutting a given circle so that the intercepted chord shall be equal to a given straight line. 18. Find the locus of the centre of a circle which touches two intersecting straight lines. 19. Find the locus of the centre of a circle which touches two parallel straight lines. 20. Describe about a circle the least possible parallelogram. 21. Inscribe in a circle the greatest possible rectangle. 22. On the base of a segment of a circle, describe the greatest possible triangle having its vertex in the circumference. 23. Find a point such that the tangents drawn from it to two given circles shall be equal to two given straight lines. 24. In a given circle draw a diameter at a given distance from a given point. State the limits of the problem. 25. If two straight lines touch a circle, the straight line bisecting the angle between them shall pass through the centre. 26. Describe a circle equal to a given circle, and touching two given straight lines. Is this problem always possible? 27. A circle touches two sides AB, AC of a triangle in F and E. Also BC is equal to BF and CE together : prove that BC touches the circle. 28. If a tangent be drawn to a circle from any point in the circumference equal to the radius, prove that the line joining its extremity with the centre is equal to the parallel chord drawn through the point of contact. . 29. If two circles touch each other externally, describe a circle which shall touch one of them in a given point, and also touch the other. In what case does this become impossible? 30. Let one circle touch another internally, and let straight lines touch the inner circle and be terminated by the outer; shew that the greatest of these lines is the one parallel to the common tangent at the point of contact. 31. If two circles touch each other externally and any third circle touch both, prove that the difference of the distance of the centre of the third circle from the centres of the other two is constant. 32. Describe a circle passing through a given point and touching a given circle in a given point. 33. Describe a circle passing through two given points and touching a given circle. 34. Describe a circle touching a given straight line in a given point, and also a given circle. 35. If three circles touch each other externally, prove that the tangents at their points of contact pass through one point. 36. Draw lines from two given points which shall meet in a given straight line and contain a right angle. Within what limits is the problem possible ? 37. In a given segment inscribe a similar segment. 38. Divide a circle into two segments such that the angle in one of them shall be double of the angle in the other. 39. Divide a circle into two segments such that the angle in one of them shall be three times the angle in the other. 40. Divide a circle into two segments such that the angle in one of them shall be five times that in the other. 41. Given the centre of a circle: describe a second circle whose radius shall be equal to the diameter of the first by means of the compasses alone. 42. Of all triangles on the same base and having their vertical angles equal to a given angle, find that which has the greatest area. 43. Parallel chords in a circle intercept equal arcs. 44. The straight lines joining the extremities of equal arcs in a circle will form a quadrilateral, two of whose opposite sides are parallel, the other two being equal, and its diagonals equal. 45. If a circle be described passing through the opposite angles of a parallelogram and cutting the four sides, and the points of intersection joined so as to form a hexagon, the straight lines thus drawn shall be parallel. 46. If two chords of a circle intersect at right angles, the opposite segments are together equal to a semicircle. 47. If two equal chords of a circle intersect at right angles, two of the segments are quadrants. 48. The diameter of a circle cuts a chord at an angle equal to half a right angle. Prove that the sum of the squares of the segments of the chord is equal to half the square on the diameter. 49. If the circumference of one circle passes through the centre of another, the lines joining the points of intersection to any point in the aforesaid circumference shall be equally inclined to the straight line joining that point and the aforesaid centre. 50. If the circumference of one circle passes through the centre of another, any two chords of the second drawn from the points of intersection so as to cut one another in the aforesaid circumference shall be equal. . |