4 BOOK OF MATHEMATICAL PROBLEMS. 



21. If a circle touch each of two other circles the indefinite 

 straight line passing through the points of contact will cut off 

 similar segments from the two circles. 



22. Two circles have internal contact at A, a straight line 

 touches one circle at P and cuts the other in Q, Q' : prove that 

 QP, PQ* subtend equal angles at A. 



If the contact be external, PA bisects the external angle 

 between QA, Q'A. 



23. A straight line touches one of two fixed circles which 

 do not intersect in P, and cuts the other in Q, Q' : prove that 

 there are two fixed points at either of which PQ, PQ' subtend 

 uiigles equal or supplementary. 



24. At two fixed points A, B are drawn AC, BD at right 

 angles to AB and on the same side of it, and of such magnitude 

 that the rectangle AC, BD is equal to the square on AB : prove 

 that the circles whose diameters are AC, BD will touch each 

 other. Prove also that the point of contact lies on a fixed circle. 



25. A triangle is inscribed in one of two concentric circles, 

 and from any point on the other circle perpendiculars are let fall 

 on the sides of the triangle : prove that the area of the triangle 

 formed by joining the feet of these perpendiculars is independent 

 of the position of the point. 



26. ABC is an isosceles triangle right angled at C, and the 

 parallelogram A BCD is completed; with centre D and distance 

 DC a circle is described: prove that if P be any point on this 

 circle, the squares on PA, PC are together equal to the square on 

 PB. 



27. A circle is described about a triangle ABC, and the tan- 

 gents to the circle at B, C meet in A' ; through A' is drawn a 

 straight line meeting AC, AB in the points K, C' : prove that 

 BB', CC' will intersect on the circle. 



