>K OF MATHEMATICAL PROBLEMS. 



7. Determine a triangle in which are given a side a, the 

 opposite angle A y and the rectangle m* under the other two sidus : 



and prove that no such triangle exists if 2m sin - > a. 



348. A triangle A'J?C f has its angles respectively supple- 

 mentary to the half angles of the triangle ABC ; and its side 

 ' equal to BC ; prove that 



. A 

 Q 2" 



. C 

 2 sin sin 



349. If x, y, z be perpendiculars from the angular points 011 

 any straight line ; prove that 



any perpendicular being reckoned negative which is drawn in the 

 opposite direction to the other two. 



350. If p lt p a , p a be the perpendiculars of the triangle, 



1111 cosA cosB costf 1 

 + + = -, - + + - ==-. 

 Pi P* P* * Ft P a P 3 -B 



351. The distances between the centres of the escribed 

 circles being respectively a, ft y ; prove that 



472 = 



y y' 







2j<r(<r-*)(<r-P)(<r-y)' 



where 2<r = a + ft + y. Prove also that 



r. (r,,+ r.,) r,r r. 



g ^ =g / =g> andA = - 



