226 BOOK OF MATHEMATICAL PROBLEMS. 



1082. Any point is joined to the angular points of a tetra- 

 hedron ABCD, and the joining lines meet the opposite faces in 

 a, 6, c, d : prove that 



Oa Ob Oc Od__, 

 I^ + M + Cc + Ud~ ' 



regard being had to the signs of the segments. Hence prove that 

 the reciprocals of the radii of the eight spheres which can be drawn 

 to touch the faces of the tetrahedron are the eight positive values 

 of the expression 



1111 



=t fc _; 

 Pi P 2 P* P* 



Pi* Pa* Pa> P* being ^h* perpendiculars from the angular points on 

 the opposite faces. 



1083. If A, , C, D be the areas of the faces of a tetra- 

 hedron ; a, b, c, a, J3, y, the cosines of the dihedral angles 

 (CM), (AB), (DA\ (DB) y (DC), respectively; then will 



_ a _ b a - c a - 2o.bc 



1084. With the same notation as in the last question, prove 

 that for all real values of /, m, n, r, 



l* + m* + n* + r*> 2mna + 2nl/3 + 2lmy + 2lra + 2mrb + 2nrc ; 



except when 



I m n r 



1085. Three straight lines are drawn, two and two at right 

 angles, through a given point, and two of them lie respectively in 

 two fixed planes : the locus of the third is a cone of the second 

 degree, whose sections parallel to the fixed planes are circles. 



1086. A point is taken within a tetrahedron A BCD so as 

 to be the centre of gravity of the feet of the perpendiculars let 



