Calendar: 1871-1872 Page 530

530 GENERAL LITERATURE AND SCIENCE straight line can be drawn from the extremity between that straight line and the circumference so as not to cut the circle 10 In circle the angle in semicircle is right angle but the angle in segment greater than semicircle is less than right angle and the angle in segment less than semicircle is greater than right angle The circles described on the equal sides of an isosceles triangle as diameters will intersect at the middle point of the base 11 If from any point without circle two straight lines be drawn one of which cuts the circle and the other touches it the rectangle contained by the whole line which cuts the circle and the part of it without the circle shall be equal to the square on the line which touches it Two circles intersect show that their common chord produced bisects their common tangent 12 To describe an isoceles triangle having each of the angles at the base double of the third angle Show that in the figure there are two triangles which possess the required property Show that there is also an isosceles triangle whose equal angles are each one-third part of the third angle 13 To inscribe an equilateral and equiangular hexagon in given circle Prove that the area of regular hexagon is greater than that of an equilateral triangle of the same perimeter 14 Triangles and parallelograms of the same altitude are to one another as to their bases Perpendiculars are drawn from any point within an equi- lateral triangle on the three sides show that their sum is invariable 15 Similar triangles are to one another in the duplicate ratio of their homologous sides 16 The rectangle contained by the diagonals of quadri- lateral figure inscribed in circle is equal to both the rectangles contained by its opposite sides 17 The straight lines drawn at right angles to the side of