Calendar: 1881-1882 Page 640

GENERAL LITERATURE AND SCIENCE 641 The sum of the squares on the sides of triangle is equal twice the square on half the base together with twice the quare on the straight line which joins the vertex to the middle point of the base The opposite angles of any quadrilateral figure inscribed in circle are together equal to two right angles Prove the converse of this theorem Four circles are described so that each one touches in- ternally three sides of quadrilateral show that circle may be described to pass through the centres of the four circles 10 To describe an isosceles triangle having each of the angles at the base double of the third angle 11 In right-angled triangle if perpendicular be drawn from the right angle to the base the triangles on either side of it are similar to the whole triangle and to one another 12 If two sides of triangle are unequal the sum of the greater side and the perpendicular on it from the opposite angle is greater than the sum of the less side and the perpen- dicular on it from the opposite angle 13 The perpendiculars drawn from the angles of triangle on the opposite sides meet at point 14 Express the area of triangle in terms of the co- ordinates of its angular points Draw the lines 2x χ and find the area of the triangle they form 15 Show that the diagonals of the parallelogram formed by the straight lines -9 are at right angles 16 Find the equation to circle of radius referred to two tangents inclined at an angle to as axes 17 If the tangent to the parabola at any point Ρ intersect the directrix in the point then SZ will be at right angles to SP