26. To make an isosceles triangle, with a given vertical angle, equal to a given triangle. 27. Let P, Q be points in AB, and AB produced, so that AP : PB :: AQ : QB, and let O be the middle point of PQ; prove AO B0= OP?. 28. In any triangle ABC the rectangle ABX AC is equal to the rectangle contained by the diameter of the circle circumscribing the triangle, and the perpendicular from A on BC. 29. Hence shew that if A be the area of a triangle ABC, D the diameter of the circumscribing circle, AxD=1 AB - BCR CA. 30. Construct a rectangle equal to a given square, and having the sum of its adjacent sides equal to a given straight line. 31. Construct a rectangle equal to a given square, and having the difference of its adjacent sides equal to a given square. 32. Describe a rectangle equal to a given square, and having its sides in a given ratio. 33. If ABC is a triangle inscribed in a circle, and the tangent at A meets BC produced in D, prove that CD : BD :: CA : BAP. 34. AB is a diameter of a circle, and at A and B tangents are drawn to the circle. If PCQ be a tangent at any point C, cutting the tangents at A, B in P, Q, prove that the radius of the circle is a mean proportional between the segments PC, QC. 35. With the same figure prove that if AQ, BP intersect in R, then CR is parallel to AP or BQ. 36. If two triangles AEF, ABC have a common angle A, prove that triangle AEF : triangle ABC= AE. AF : AB. AC. 37. Given two points in a terminated straight line, find a point in the straight line such that its distances from the extremities of the line are to one another in the same ratio as its distances from the fixed points. 38. Divide a given straight line into two parts such that their squares may have a given ratio to one another. 39. AB is divided in C; shew that the perpendiculars from A, B on any straight line through C have to one another a constant ratio. 40. From the obtuse angle of a triangle to draw a line to the base which shall be a mean proportional between the segments of the base. 41. Divide a given triangle into two parts which shall have to one another a given ratio by a line parallel to one of the sides. 42. If from any point in the circumference of a circle perpendiculars be drawn to the sides, or sides produced of an inscribed triangle, prove that the feet of these perpendiculars lie in one straight line. . 43. If a line be divided into any two parts to find the locus of the point in which these parts subtend equal angles. 44. If two circles touch each other externally, and also touch a straight line, prove that the part of the line between the points of contact is a mean proportional between the diameters of the circles. 45. Any regular polygon inscribed in a circle is a mean proportional between the inscribed and circumscribed regular polygons of half the number of sides. 46. ABC is a triangle, and O is the point of intersection of the perpendicular from ABC on the opposite sides of the triangle: the circle which passes through the middle points of OA, OB, OC, will pass through the feet of the perpendiculars, and through the middle points of the sides of the triangle. 47. Describe a circle to touch a given straight line and a given circle, and to pass through a given point. 48. A and B are two points on the same side of a straight line which meet AB produced in C. Of all the points in this straight line find that at which AB subtends the greatest angle. 49. Inscribe a square in a given pentagon. 50. ABCD is a quadrilateral figure circumscribing a circle, and through the centre 0, a line EOF equally inclined to AB and BC is drawn to meet them in E and F: prove that AE : EB :: CF: FD. CAMBRIDGE: PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS. |