Calendar: 1855-1856 Page 370

3G6 GENERAL LITERATURE AND SCIENCE JUNIOR MATHEMATICAL SCHOLARSHIP JrucIttJ anfc Crigonomctrp In any right-angled triangle the square which is described upon the side subtending the right angle is equal to the squares described upon the sides which contain the right angle Prove that- GH FD2 KE2 AB2 AC2 BC2 To describe an isosceles triangle having each of the angles at the base double of the third angle If Β be produced to meet the circle in Ε and and 25 be joined then Δ CEF Α Α Β and Δ Β is mean proportional between CEF and BCD In every triangle the square of the side subtending either of the acute angles is less than the squares of the sides containing that angle by twice the rectangle contained by either of these sides and the straight line intercepted between the perpendicular let fall upon it from the opposite angle and the acute angle From the vertex of triangle Β draw AD ABC or Β produced from Β draw Β Ε bisecting in then shall BE CEP Β Β If from point without circle there be drawn two straight lines one of which cuts the circle and the other meets it if the rectangle contained by the whole line which cuts the circle and the part of it without the circle be equal to the square of the line which meets it the line which meets shall touch the circle Β is an equilateral triangle the centre of the inscribed circle through Ο draw AEOD meeting the circle in Ε and the base in AD shall be bisected in and If two triangles have one angle of the one equal to one angle of the other and the sides about the equal angles propor- tionals the triangles shall he equiangular and shall have those angles equal which are opposite to the homologous sides