about a circle are equal; and the angles, subtended at the centre by any two opposite sides, are together equal to two right angles. 21. From a point A without a circle, two tangents AB and AC are drawn, and at a point F, on the arc BFC, between the two points of contact B and C, another tangent EFD is drawn, meeting AB, AC in the points E and D; it is required to shew that the perimeter of the triangle AED is invariable, at whatever point of the arc the tangent EFD is drawn; also that the angle subtended at the centre by this tangent is invariable. 22. Two chords of a circle being given in magnitude and position, it is required to describe the circle. 23. If the adjacent extremities of two intersecting chords be joined, the triangles so formed will be equiangular. 24. If two secants from the same point make equal angles with a line drawn to the centre from that point, the parts of the secants within the circle are equal. 25. In the diameter of a circle produced, determine a point from which to draw a tangent that shall be equal, (1) to the diameter; (2) to a given line. 26. Given the base, the vertical angle, and the altitude of a triangle, to construct the triangle. 27. If from any two points in the circumference of the greater of two concentric circles tangents be drawn to the lesser circle, they shall be equal to one another. 28. If two circles touch internally or externally, two straight lines drawn through the point of contact will intercept arcs, the chords of which are parallel. 29. If two circles touch internally, and the radius of the one be equal to the diameter of the other, find the locus of the middle point of any straight line drawn from the point of contact to the outer circumference. 30. If from the centre of a circle a straight line be drawn to any point in a given chord, the square of that line, together with the rectangle of the segments of the chord, is equal to the square of the radius. 31. If two chords in a circle cut one another at right angles, the sum of the square of the segments is equal to the square of the diameter. 32. To draw a tangent to a circle, (1) which shall be parallel to a given straight line; (2) which shall make a given angle with a given line. 33. A diameter of a circle is produced to a given point. Find a point between it and the circumference, such that a tangent from that point shall be equal to the segment of the line between the two points. 34. Through a given point P in a given circle, to draw a straight line AB, which, terminating in the circumference, shall be equal to a given line. When is the problem impossible? 35. Through a given point P in a given circle, to draw a straight dine AB, meeting the circumference in A and B, such that AP and BP may have a given difference. 36. If two circles touch one another externally or internally, a straight line drawn through the point of contact will form, with the diameters and lines joining the corresponding extremities, triangles which shall be equiangular. 37. The square of the side of an equilateral triangle, inscribed in a circle, is equal to three times the square of the radius. 38. If from any point in the diameter of a semicircle, two straight lines be drawn to the circumference, one to the middle of the arc, the other perpendicular to the diameter, the sum of their squares shall be double the square of the radius. 39. If two circles cut one another, and the points of intersection be joined, two tangents drawn from any point in this line produced will be equal to one another; and conversely, if the tangents be equal the line produced will pass through the point of concourse of the tangents. 40. The straight line drawn from one angle of an equilateral triangle, inscribed in a circle, to any point in the opposite circumference, is equal to the sum of the lines drawn from the other two angles to the same point. 41. If PA and PB be two tangents drawn from the same point P, without a circle, and if PB be produced to meet in D, a diameter drawn from A, and if the points of tangency A, B be joined, it is required to shew that the angle P, contained by the tangents, is double of the angle at A between the diameter and the line AB, joining the points of contact. 42. If an equilateral triangle be inscribed in a circle, and the adjacent arcs, cut off by two of its sides, be bisected, the line joining the points of bisection will be trisected by the sides (Vide Ex. 13). 43. The angle subtended at the centre of a circle by the part of a tangent, intercepted between two other tangents, drawn from the extremities of a diameter, is a right angle. 44. AB is the common chord of two intersecting circles, and P any point in it; two chords OPE and FPL are drawn, one in each circle; shew that the locus of the four extremities, O, L, E, F, is the circumference of a circle. 45. ABC is a triangle inscribed in a circle OABC; perpendiculars OL, OE are drawn to the sides AB, AC, and OF to BC produced through C; shew that the extremities of these perpendiculars, F, E, L, are in the same straight line. 46. If two circles touch each internally in A, and two lines ApP, AqQ be drawn through A, cutting the circles in p, P, and q, Q, prove that the line pq is parallel to the line PQ. 47. Two circles touch each other internally in A, a chord PQ of the outer touches the inner in C, prove that AC bisects the angle PAQ. 48. Describe a circle- (1) Which shall touch two given lines, and have its radius equal to a given line; if the lines are parallel, when is the problem impossible? (2) Touching a given line in a given point, and touching a given circle. (3) Touching a given circle in a given point, and touching a problem has three cases, consider each case separately. (9) Touching two given circles, and passing through a given point. (10) Touching three given circles; consider the four cases. 58. Through a given point draw a straight line cutting a circle, so that the part within the circle may be equal to a given line. When is the problem impossible? 59. Through a given point without a circle, to draw a line cutting the circle, so that the part between the point and the circumference may be equal to a given line. When is the problem impossible? 60. Two circles touch each other externally; describe a circle which shall touch one of them in a given point, and also touch the other. When is the problem impossible? BOOK IV. DEFINITIONS. 1. A rectilineal figure is said to be inscribed in another rectilineal figure when all the angles of the one are on the sides of the other. 2. A figure is said to be described about, or to circumscribe another figure, when all its sides pass through the angular points of the other. 3. A rectilineal figure is said to be inscribed in a circle when all its angles are on the circumference of the circle. 4. A rectilineal figure is said to be described about, or to circumscribe a circle, when each of its sides touches the circumference of the circle. 5. A circle is said to be inscribed in a rectilineal figure, when the circumference touches each side of the figure. PROPOSITION. I.-PROBLEM. (Etc. IV. 1.) In a circle to inscribe a chord equal to a given straight line. Let SPT be the given circle, and M the given line, it is required to inscribe within the circle a chord equal to M. M is greater than the diameter, the problem (III. 2, cor.) is evidently impossible. Q. E. F. PROP. II.-PROBLEM. (EUC. IV. 2.) In a given circle to inscribe a triangle equiangular to a given triangle. Let PQS be the given circle, and ABC the given triangle, Join S, Q. Then the triangles SPQ and ABC are equi angular. For since QP cuts the circle, and DE touches it in the same point P, the angle QPE is equal to the angle S in the alternate segment. But QPE is equal to the angle B, therefore the angle S is also equal to the angle B. (Const.) (III. 12, cor. 3.) (Const.) (Ax. 1.) |