10 BOOK OF MATHEMATICAL PROBLEMS. 



G7. A point is taken within a triangle ABC, and through 

 A y B, Care drawn straight lines parallel to those bisecting the 

 angles BOG, CO A, AOB : prove that these lines will meet in a 

 point. 



68. ABC is a triangle, AA', BB 1 , CO' are drawn through a 

 point to meet the opposite sides : prove that the straight lines 

 drawn through A, B, C to bisect B'C', C'A', A'B' will meet in a 

 point. 



69. If two circles lie entirely without each other, and any 

 straight line meet them in P, P' ; Q t Q' respectively, there are two 

 points such that the straight lines bisecting the angles POP, 

 QOQ' shall be always at right angles to each other. 



70. Given two circles which do not intersect, a tangent to 

 one at any point P meets the polar of P with respect to the other 

 in P : prove that the circle on PP f as diameter will pass through 

 two fixed points. 



71. A point has the same polar with respect to each of two 

 circles, prove that any common tangent will subtend a right 

 angle at that point. 



72. Given two points A,B; if any straight line PA Q be drawn 

 through A so that the angle PBQ is equal to a given angle, and 

 that BP has to BQ a given ratio, P, Q will lie on two fixed circles 

 which pass through A and B. 



73. If be a fixed point, P any point on a fixed circle, and 

 the rectangle be constructed of which OP is a side and the 

 tangent at P a diagonal, the angular point opposite will lie on 

 the polar of 0. 



