BOOK OF MATHEMATICAL PROBLEMS. 



50. Two circles intersect in A, B, any straight line through 

 A meets the circles in P, Q: prove that BP has to BQ a constant 

 ratio. 



51. Through the centre of perpendiculars of a triangle is 

 drawn a straight line at right angles to the plane of the triangle : 

 prove that any tetrahedron of which the triangle is one face, and 

 whose opposite vertex lies on this line will be such, that through 

 any edge can be drawn a plane perpendicular to the opposite 

 edge. 



52. ABCD are four points not in one plane, and AB, AC 

 respectively lie in planes perpendicular to CD, BD : prove that 

 AD lies in a plane perpendicular to BC ; and that the middle 

 points of the six edges lie on a sphere, which will also pass 

 through the feet of the shortest distances between the opposite 

 edges. 



53. Each edge of a tetrahedron is equal to the opposite edge : 

 prove that the straight line joining the middle points of two 

 opposite edges is at right angles to both. 



54. If from any point be let fall perpendiculars Oa, Ob, 

 Oc y Od on the faces of a tetrahedron ABCD, the perpendiculars 

 from A, B, C, D on the corresponding faces of the tetrahedron 

 abed meet in a point 0' ; and the relation between and 0' is 

 reciprocal. 



55. A solid angle is contained by three plane angles : prove 

 that any straight line through the containing point makes with 

 the edges angles whose sum is greater than half the sum of the 

 containing angles. 



56. The circles described on the diagonals of a complete 

 quadrilateral as diameters cut orthogonally the circle circumscribing 

 the triangle formed by the diagonals. 



57. Four points are taken on the circumference of a circle, 

 and through them are drawn three pairs of straight lines each 

 intersecting in a point : prove that the straight line joining any one 



