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
