78 BOOK OF MATHEMATICAL PROBLEMS. 



3G9. The sides of a rectangle are 2a, 26, and the angles sub- 

 tended by its diagonals, at a point whose distance from the centre 

 is c t are a, ft ; prove that 



/f ( f +6 ?' = a 9 (tan a tan ft)*+ b* (tan a * tan ft)'. 

 (a + o c ) 



370. The diagonals 2a, 26 of a rhombus subtend angles a, ft 

 at a point whose distance from the centre is c' } prove that 



(a* - c 2 ) 8 b' tan 2 a + (6 8 - c 8 ) 2 a 2 tan 8 )8 = 4 S 6 V. 



371. Three circles A, B, C touch each other two and two, 

 and one common tangent to A and B is parallel to one common 

 tangent to A and C ; prove that, if a, 6, c be their radii, and p, q 

 the distances of the centres of B and C from that diameter 'of A 

 which is perpendicular to the two parallel tangents, 



pq = 2a* = Sbc. 



372. AB is the diameter of a circle, C any point on AB, on 

 AC, BC as diameters are described two other circles : if a circle 

 be described touching the three, its diameter will be equal to the 

 distance of its centre from AB. 



373. Four points A, P, Q, B lie in a straight line, circles are 

 described on AQ(= 2a), BP (= 26), and AB (= 2c) as diameters ; 

 prove that the radius of a circle touching the three is 



c (c - a) (c - b) 

 c*-ab 



374. A polygon of n sides inscribed in a circle is such that 

 its sides subtend angles a, 2a, ... no. at the centre; prove that its 

 area is to the area of the regular inscribed polygon of n sides in 

 the ratio 



. no. . a 



sin -jj- : n sm ^ . 



375. A BCD is a parallelogram and P any point within it ; 

 prove that 



&APC cot APC ~ A BPD cot BPD 

 is independent of the position of P. 



