GEO.Mr.lKV. 



of these points to the centre of the circle will be perpendicular to 

 the line joining the other two. 



58. A sphere is described touching three given spheres: 

 prove that the. plane passing through the points of contact, contains 

 one of four fixed straight lines. 



. Four straight lines are given in position: prove that an 

 infinite number of systems of three circles can be found, such that 

 tin joints of intersection of the four straight lines are the cv 

 uilarity of the circles taken two and two. 



60. In two fixed circles are drawn two parallel chords PI'', 

 QQ ' PQ, P'Q are joined meeting the circles again in /?, S\ :. 

 respectively : prove that the points of intersection of QQ' 9 /?/f and 

 f /'/', *S'*S" lie on a fixed straight line. 



61. The six radical axes of the four circles (taken two 

 two) which touch the sides of a triangle are the straight HIM < 

 bisecting internally and externally the angles of a triangle f- ; 



by j-'ining the middle points of the sides of the former triangle. 



If two circles have four common tangent 



described on these common tangents as diameters will have a 

 common radical axis. 



63. Four points are taken on a circle, and from tin- middl.> 

 of tin- chord joining any two a straight linr is drawn per- 

 Ml ,ir to the choi.l joining the other two: prove that the six 

 lines so drawn will meet in a point. 



Given in position two sides of a triangle, im-1 

 angle equal to that of an equilateral triangle, and giv.-n tl, length 



third side : prove that the centre of the Nine Points' ( 

 of th triangle lies on a fixed straight ]in<>. 



65. Given in position two sides of a triangle, and given 

 sum ;' those sides, the centre of the Nine Point*' Circle lies on a 



* straight 1 



66. The perpendiculars 1- centres of the escribed 



.:1-- on the corresponding sides will n 



