Calendar: 1886-1887 Page 685

GENERAL LITERATURE AND SCIENCE 687 given circle such that the part of it intercepted by the circle may be equal to given chord of the circle 10 The angles in the same segment of circle are equal to one another On given straight line describe triangle whose vertical angle is equal to given angle Prove that there are an infinite number of such triangles and that the centres of the circles inscribed in them all lie on fixed circle 11 From given circle cut off segment which shall contain an angle equal to given rectilineal angle Inscribe in circle triangle which shall be equiangular with given triangle 12 Inscribe circle in given triangle How many circles can be described touching the sides or the sides produced of given triangle Prove that the straight line joining the centres of any two such circles passes through one of the angular points of the triangle 13 Describe an isosceles triangle having each of the angles at the base double the third angle If ABC is such triangle being the vertex and if the bisectors of the angles at the base meet one another in and meet the circle circumscribing the triangle in Ε and prove that AEDF is rhombus 14 Similar triangles are to one another in the duplicate ratio of their homologous sides If Α'ΒΌ' are three points in the sides AB BC CA respectively of triangle ABC such that ΑΑ' AB BB1 BC CC iCA prove that the triangle A'B'C will be of the triangle ABC 15 If two straight lines be cut by parallel planes they shall be cut in the same ratio If AB CD are any two equal straight lines in space not in the same plane prove that the middle points of the four straight lines AC AD BC BD will be the angular points of rhombus