PROBLEMS. BOOK IV. 1. In a given circle place a line of given length, which shall pass through a given point. 2. Draw that diameter of a circle which shall pass at a given distance from a given point. 3. The square of the side of an equilateral triangle inscribed in a circle is triple the square of the side of the regular hexagon inscribed in the same circle. 4. An equilateral triangle is inscribed in a circle, and through the angular points tangents are drawn: shew that they will form an equilateral triangle, whose area is four times the former. 5. Shew that a circle may be described through the centres of four circles, each of which touches one side and the two adjacent sides produced of any quadrilateral. 6. If two circles be described, one without, the other within, a right-angled triangle, the sum of their diameters is equal to the sum of the sides containing the right angle. 7. The perpendicular from the vertex on the base of an equilateral triangle is equal to the side of an equilateral triangle inscribed in a circle whose diameter is the base. 8. Inscribe a square and a circle in a given quadrant. 9. Shew how to find the centres of the escribed circles of a triangle; and prove that the line joining any two of these centres is perpendicular to the line joining the centre of the inscribed circle with the angular point between them. 10. The line bisecting any angle of a triangle inscribed in a circle cuts the circumference in a point, which is equidistant from the extremities of the side opposite to the bisected angle and from the centre of the inscribed circle. 11. ABCD is a rectangle: if in the triangle ABC a circle be inscribed, touching AB, BC in E, F, and EGH, FGK be drawn parallel to AD, CD, then the rectangle KH is equal to the gnomon AFH. 12. The locus of the centres of the circles inscribed in all right-angled triangles on the same hypothenuse is the quadrant described on the hypothenuse. 13. In any triangle ABC, if AD, bisecting the angle A, cut BC in D, and from O, the centre of the inscribed circle, OE be drawn perpendicular to BC, shew that the angles BOE, COD are equal. 14. Inscribe a square in a given right-angled isosceles triangle. 15. Through two given points describe a circle touching a given circle: and shew that, of all lines that can be drawn from the two points to meet the convex circumference, those drawn to the points of contact thus obtained will contain the greatest possible angle. 16. The area of an inscribed regular hexagon is that of the one circumscribed about the same circle. 17. Upon a given line as diagonal describe a rhombus, so that two of its angles shall be double of the other two. Hence shew how a right angle may be trisected. 18. Given the vertical angle, the line drawn to the base bisecting that angle, and the difference between the base and the sum of the sides: construct the triangle. 19. Draw from the obtuse angle A of a given triangle to the base a line whose square shall be equal to the rectangle of the segments of the base. 20. The centres of the inscribed and circumscribed circles of an equilateral triangle coincide, and the diameter of one is double that of the other. 21. The line joining the centres of the inscribed and circumscribed circles of a triangle subtends at any one of the angular points an angle equal to the semidifference of the other two angles. 22. Given the angles of a triangle and the radius of the inscribed circle: construct the triangle. 23. Given the vertical angle of a triangle, and the radii of the inscribed and circumscribed circles: construct the triangle. 24. Within a given circle place eight circles, touching each other and the given circle. 25. The lines joining the alternate angles, or the intersections of the alternate sides of a regular pentagon, will form another regular pentagon. 26. Upon a given straight line describe a regular octagon. 27. Inscribe in a given circle a rectangle equal to a given rectilineal figure. 28. The area of a regular octagon inscribed in a circle is equal to the area of the rectangle of the sides of the inscribed and circumscribed squares. 29. Inscribe a square in the space included between two equal circles which cut each other. 30. Inscribe a circle in a given rhombus. 31. ABCD is a quadrilateral inscribed in a circle: produce AD, BC to meet in E; from any point F in DE draw FH parallel to BE meeting CD in H, and join FB cutting the circle in G: then shew that GH will cut the circle again in a fixed point K. 32. Inscribe an equilateral triangle in a given square, when the vertex is (i) in the middle of a side, (ii) in one angle. 33. Inscribe the least possible square in a given square. 34. Describe a circle which shall pass through one angle and touch two sides of a given square. 35. In the figure (4. 10), shew that AC is the side of a regular decagon inscribed in the larger circle, and of a regular pentagon inscribed in the smaller. 36. In the figure (4. 10) produce DC to meet the circle in F: then the angle ABF shall be triple of the angle BFD. 37. If ABCDE be a regular pentagon, shew that the angles ABE, BCA, CDB, DEC, EAD, are together equal to two right angles. 38. Given a regular pentagon: describe a triangle of the same area and altitude. 39. If two diagonals of a regular pentagon be drawn cutting one another, the larger segments will be each equal to a side of the pentagon. 40. Divide a right angle into five equal parts. 41. The opposite sides of a regular hexagon are parallel and if any two sides of an inscribed hexagon are parallel to two other sides, the remaining two will also be parallel. 42. Inscribe a regular hexagon in a given equilateral triangle, and compare its area with that of the triangle. 43. Inscribe a regular dodecagon in a given circle, and shew that its area equals that of a square on the side of an equilateral triangle inscribed in the same circle. 44. Let AB, CD, two alternate sides of a regular polygon, be produced to meet in E: shew that the figure AECO can be inscribed in a circle, being the centre of the polygon. 45. Given a regular polygon inscribed in a circle: shew how to describe another, with double the number of sides, in or about the same circle. 46. If R, r be the radii of circles described about and in a regular polygon, and R', r' the corresponding radii for a regular polygon of same perimeter and twice the number of sides, shew that r' = (R + r), and R2 = Rr'. 47. If from any point within a regular polygon of n sides perpendiculars be dropped on the sides, their sum will be n times the radius of the inscribed circle. How may this statement be made to include the case when the point is without the polygon? 48. If from the angles of an equilateral triangle, parallel lines, or ordinates, be drawn to any given line without it, their sum, and also the sum of the abscissæ, or segments of the line cut off by them, measured from a fixed point in it without them all, will be respectively three times the ordinate and abscissa of the centre of the inscribed circle. Shew that this is true of any regular polygon; and explain how the statement may be made to include the case of the line cutting the figure, and of the fixed point being any whatever in it. 49. If from the angles of an equilateral triangle ordinates be drawn to the diameter of the circumscribing circle, the single ordinate which falls on one side of the diameter will equal the sum of the two which fall on the other side; and so also the single segment of the diameter on one side of the centre will equal the sum of the two upon the other side. 50. If from any point P lines be drawn to the angles of an equilateral triangle ABC, shew that PÃ2 + PB2 + PC2 is constant: and hence if ABCP, A'B'C'P' be concentric circles, and ABC, A'B'C' equilateral triangles inscribed in them, shew that AP'2+ BP'2 + CPA'P2 + B ́P2 + C'P2. |