PARALLEL FORCES 99 



4. Forces P, Q, E act along the sides of a triangle ABC, and their 

 resultant passes through the centers of the inscribed and circumscribed 

 circles. Prove that 



P = Q = R 



cos B cos C cos C cos A cos A cos B 



5. If four forces acting along the sides of a quadrilateral are in equi- 

 librium, prove that the quadrilateral must be plane. 



6. ABCD is a plane quadrilateral and forces represented by AB, CJ5, CD, 

 AD act along these sides, respectively, of the quadrilateral. Show that if there 

 is equilibrium, the quadrilateral must be a parallelogram. 



7. If a quadrilateral can be inscribed in a circle, prove that forces acting 

 along the four sides and proportional to the opposite sides will keep it in equi- 

 librium. Show also that the converse is true, namely that for equilibrium, the 

 forces must be proportional to the opposite sides. 



8. A quadrilateral is inscribed in a circle, and four forces act along the 

 sides, and are inversely proportional to the lengths of these sides. Show that 

 the resultant has for line of action the line through the intersections of pairs 

 of opposite sides. 



9. Forces act along the four sides of a quadrilateral, equal respectively to 

 a, 6, c, and d times the lengths of those sides. Show that if there is equilibrium, 



ac = bd, 



and that the further conditions necessary to insure equilibrium are that the 

 ratios a : 6 and 6 : c shall be the ratios in which the diagonals are divided at 

 their points of intersection. 



10. In the last question show that the perpendicular distances to the first 

 side, from the two points of the quadrilateral which are not on that side, are in 



the ratio 



a(c-6) :d(b-a). 



PARALLEL FORCES 



74. Let us use the method just explained, to determine the 

 resultant of two parallel forces P, Q. 



Take any point on the line of action of P as origin, and take 

 this line of action of P for axis Oy, as in fig. 53. Let the resultant 

 be R, components X, Y. Then resolving we obtain 



X= 0, 

 F-P + ft 



so that the resultant force is of magnitude P + Q and acts parallel 



