7 '2 BOOK OP MATHEMATICAL PROBLEMS. 



334. In the side -6(7, produced if necessary, find a point P 

 such that the square on PA may be equal to the sum of the 

 squares on PB, PC j and prove that this is only possible when 

 A, B, C are all acute and tan A < tan B + tan (7, or when B or G 

 is obtuse. These conditions being satisfied, prove that there are 

 in general two such points which lie both between B afcd (7, one 

 between and one beyond, or both beyond, according as A is the 

 greatest, the mean, or the least angle of the triangle. 



335. P, Q are two points on the circumscribed circle, the 

 distance of either from A being a mean proportional between its 

 distances from B and C ; prove that 



336. The line joining the middle points of BG and of the 

 perpendicular from A on BG makes with BC an angle 



cot' 1 (cot B ~ cot C). 



337. The line joining the centres of the inscribed and cir- 

 cumscribed circles makes with BC an angle 



j ( sin B ~ sin C 

 \ cos B + cos C- 



338. The line joining the centre of the circumscribed circle 

 and the centre of perpendiculars makes with BC an angle 



_. ( tan B ~ tan G 

 cot 



339. The perpendicular from A on BC is a harmonic mean 

 between r a and r. 6 . 



340. If be the centre of the circumscribed circle and A 

 meet BC in D, 



DO : AO :: cos 4 : coa(B-C). 



