44. Find the path of a billiard ball started from a given point which, after being reflected from the four sides of the table, will pass through another given point. 45. If two lines bisecting two angles of a triangle and terminated by the opposite sides be equal, the triangle is isosceles. 46. State and prove the Proposition corresponding to Exercise 41, when the base and difference of the sides are given. 47. If a square be inscribed in a triangle, the rectangle under its side and the sum of the base and altitude is equal to twice the area of the triangle. 48. If AB, AC be equal sides of an isosceles triangle, and if BD be a perpendicular on AC; prove that BC2 = 2AC.CD. 49. The sum of the equilateral triangles described on the legs of a right-angled triangle is equal to the equilateral triangle described on the hypotenuse. 50. Given the base of a triangle, the difference of the base angles, and the sum or difference of the sides; construct it. 51. Given the base of a triangle, the median that bisects the base, and the area; construct it. 52. If the diagonals AC, BD of a quadrilateral ABCD intersect in E, and be bisected in the points F, G, then 53. If squares be described on the sides of any triangle, the lines of connexion of the adjacent corners are respectively—(1) the doubles of the medians of the triangle; (2) perpendicular to them. BOOK II. THEORY OF RECTANGLES Every Proposition in the Second Book has either a square or a rectangle in its enunciation. Before commencing it the student should read the following preliminary explanations: by their assistance it will be seen that this Book, which is usually considered difficult, will be rendered not only easy, but almost intuitively evident. 1. As the linear unit is that by which we express all linear measures, so the square unit is that to which all superficial measures are referred. Again, as there are different linear units in use, such as in this country, inches, feet, yards, miles, &c., and in France, metres, and their multiples or sub-multiples, so different square units are employed. 2. A square unit is the square described on a line whose length is the linear unit. Thus a square inch is the square described on a line whose length is an inch; a square foot is the square described on a line whose length is a foot, &c. 3. If we take a linear foot, describe a square on it, divide two adjacent sides each into twelve equal parts, and draw parallels to the sides, we evidently divide the square foot into square inches; and as there will manifestly be 12 rectangular parallelograms, each containing 12 square inches, the square foot contains 144 square inches.