CONIC SECTIONS, ANALYTICAL. 157 



and (4) to the circle to which the triangle is self-conjugate 

 a 8 tan A + y* tan B + tan C = 0. 



748. If A' y B y (? be the feet of the perpendiculars let fall 

 from the point la mfi = ny on the sides of the triangle of refer- 

 ence, the straight lines drawn through A, B t C perpendicular to 

 EC', C'A' t A'B' will meet in the point 



"= ^ JL. 



/a* mb* nc* ' 



749. A triangle A'B'CT has its angular points on the sides of 

 the triangle ABC, and A A', BE, CO' meet in a point, any straight 

 line is drawn meeting the sides of the triangle A'B'C' in points 

 which are joined respectively to the corresponding angles of the 

 triangle ABC ; prove that the joining lines meet the sides of the 

 triangle ABC in three points lying in one straight line. 



750. The two points at which the escribed circles of the 

 _;le of reference subtend equal angles lie on the straight 



line 



a (b - c) cot A + ft (c - a) cot B + y (a- b) cot C = 0. 







! . Four straight lines form a quadrilateral, and from the 

 middle points of the sides of a triangle formed by three of them 

 perpendiculars are let fall on tin- line joining the middle points of 

 the diagonals ; prove that these perpendiculars are inversely pro- 

 iial to the perpendiculars from the angular points pf the 

 triangle on the fourth straight line. 



1. A straight line meets the sides of a triangle in A', K, C\ 



the straight line joining A to the point (BK, CC') meets BC in D t 



are similarly deter If be any point, the lines 



joining />, /;, /' t , the ! , ns of OA, OB, 00 



with A'B'C' will pans 0', and 0& will pass 



igh a point whose position is independent of 0. 



