PLANE TKICOXOMETRY. 73 



1. The perimeter of a triangle : perimeter of the inscribed 

 circle :: the area of the triangle : area of the circle 



A B C 



: : cot 9 cot -^ cot : TT. 



342. A triangle is formed by joining the feet of the per- 

 pendiculars of the triangle ABC', and the circle inscribed in this 

 triangle touches the sides in A', B', C' ; prove that 



VA' A'E 



343. A circle is drawn to touch the circumscribed circle and 



ides AB, AC ' t prove that its radius is rsec* ~ ' *&& ^ i* 



touch the circumscribed circle and the sides AB, AC produced, its 



radius is r sec* -r . If B = C and the latter radius = 7?, cos A = ( . 

 i 2 J 



3 14. Prove the formula*, 



6' Bin 2(7 - 2bc sin (B-C)- c f sin 25 = 0, 

 I* cos 2(7 + 2bc cos (B - C) + c f cos 2B = a 9 . 



345. Having given 



\n'C + zzin.'B = zam'A + x tin* C = x &m* B + y sin'J ; 



prove that 



I )ctcrmiue a triangle having a base c, an altitude h, mul 

 nee a of the base angles; and if 0,, 6 t be the two 



values of the vertical angle, prove that cot 0, + cot 0, = J ~i^* 



Prove that only one of these values corresponds to a true solu: 

 an.l. if this bed,, that 



, 



; X 



- c(l-cWa) 



