IOC BOOK OP MATHEMATICAL PROBLEMS. 



536. If ABC be an acute-angled triangle, P any point in its 

 plane ; the three circular loci 



PA = PB' + PC', PB'=PC* + PA, PA'=PB' + PC* 



will have their radical centre at the centre of the circle circum- 

 scribing the triangle. 



537. The radii of two circles are a, 6, and the distance 

 between their centres ^/{2 (a* + 1 3 )} ; prove that any common 

 tangent subtends a right angle at the point bisecting the distance 

 between their centres. 



538. A certain point has the same polar with respect to each 

 of two circles ; prove that any common tangent subtends a right 

 angle at that point. 



539. AS is a diameter of a circle, a point in* a fixed 

 straight line passing through A, from two tangents are drawn 

 to the circle meeting the tangent at B in P, Q ; prove that 

 BP + BQ is constant. 



540. AB is a diameter of a circle, a chord through A meets 

 the tangent at B in P, and from any point in the chord produced 

 are drawn two tangents to the circle ; prove that the lines joining 

 A to the points of contact will meet the tangent at B in points 

 equidistant from P. 



541. Three circles A, B, C have a common radical axis, and 

 from any point on C two tangents are drawn to A, B respect- 

 ively ; prove that the ratio between the squares on these tangents 

 is equal to the ratio between the distances of the centres of A, B 

 from the centre of C. 



542. On two circles are taken two points such that the 

 tangents drawn each from one point to the other circle are 

 equal ; prove that the points are equidistant from the radical 

 axis. 



543. There are two systems of circles such that any circle of 

 one system cuts any circle, of the other system at right angles ; 



