SOLID GEOMETRY. 



1077. If x, y be the lengths of two of the lines joining the 

 middle points of opposite edges of a tetrahedron, <o the angle 

 between these lines, and a, a those edges of the tetrahedron 

 which are not met by either of the lines, 



1078. The lengths of the three pairs of opposite edges of a 

 tetrahedron are , a \ b, &'; c, c: prove that, if be the acute 

 angle between the directions of a and a', 



2aa' 



1079. The line joining the centres of the two spheres which 



touch the faces of the tetrahedron A BCD opposite to A, B respec- 



tively, and the other faces produced, will intersect the edge CD 



in a point P such that CP : PD :: &ACB :&ADB-, and the 



( B (produced) in a point Q such that 



AQ : BQ :: CAD : &CBD. 



1080. On three straight lines, meeting in a point, are taken 

 point.> , 6; C, c respect i\vly : prove that tin- intersections of 

 tin- planes ABC, abc-, aBC, Abe-, AbC, aBc; and .I//.-, bC all lie 

 on one plane which divides each of the three straight lines har- 

 monically. 



1081. If through ai bo drawn three straight lines 

 each meeting two opposite edges of a tetrahedron ABCD\ and if 



ft > P> ^ y be the points where these straight lines met t 

 , A&-, CA, BD- t AB, CD; then will 



Ba.Cy. Dp~p. Ca . />y, 

 Cb.Au.Dy~Cy.Ab.Da, 



w. i:. 



