G BOOK OF MATHEMATICAL PROBLEMS. 



3G. The radii from the centre of the circumscribed circle of a 

 triangle to the angular points are respectively perpendicular to 

 the lines joining the feet of the perpendiculars. 



37. Three circles are described each passing through the 

 centre of perpendiculars of a given triangle and through two 

 of the angular points : prove that their centres are the angular 

 points of a triangle equal in all respects to the given triangle and 

 similarly situated : and that the relation between the two triangles 

 is reciprocal. 



38. If the centres of two of the circles which touch the sides 

 of a triangle be joined, and also the centres of the other two, the 

 squares on the joining Jines are together equal to the square on 

 the diameter of the circumscribed circle. 



39. The centre of perpendiculars of a triangle is joined to the 

 middle point of a side, and the joining line produced to meet the 

 circumscribed circle : prove that it will meet it in the same point 

 as the diameter through the angular point opposite to the bisected 

 side. 



40. From any point of a circle two chords are drawn touching 

 another circle whose centre is on the circumference of the former : 

 prove that the line joining the extremities of these chords is fixed 

 in direction. 



41. ABC is a triangle and the centre of its circumscribed 

 circle; A'J?C' another triangle whose sides are respectively 

 parallel to OA, OB, 00. If through A', &, 0' be drawn straight 

 lines respectively parallel to the sides of the former triangle, they 

 will intersect in a point which is the centre of one of the circles 

 touching the sides of the triangle A'B'G'. 



42. A triangle is drawn having its sides parallel to the lines 

 joining the angular points of a given triangle to the middle points 

 of the opposite sides : prove that the relation between the two 

 triangles is reciprocal. 



43. Two triangles are so related that straight lines drawn 

 through the angular points of one parallel respectively to the sides 



