CONIC SECTIONS, ANALYTICAL. 159 



758. The equation of the straight line bisecting the diagonals 

 of the quadrilateral whose four si<l s un> 



la m/3 ny = 0, 



is ?a + m*P + n'y = Q, 



and the equation of the radical axis of the three circles on the 

 diagonals is 



...... + ...... = 0. 



759. One of four straight lines passes through the centre of 

 one of the four circles which touch the diagonals of the quadrila- 

 teral ; prove that the other three straight lines pass each through 

 one >f the other three centres. Prove that in this case the circles 



!l>ed on the three diagonals touch each other in a point lying 

 on the circle circumscribing the triangle formed by the diagonals, 

 and that their common tangent is a normal to this circle. 



760. The equation of a circle passing through B and C, and 

 whose segment on BC (on the same side as A) contains an angle 



e, is 



7G1. The locus of the radical centre of three circular arcs on 

 BC, CM, J/A respectively containing angles A + 0, + 0, C + G for 

 different values of 6, is the straight line 





Km A am 2? 



If = 90, the radical centre is the centre of the circumscril < 1 

 circle. 



7G2. Prove that, if /' l.c a point such that 

 tan BPC- tan A = tan CPA - tan B = tan APB - tan C, 



are two positions of /*; and that the equation of the line 

 joining them is 



a cot .1 (tan B - tan C) + ft cot B (tan C- tan A) 



+ y cot C (tan A - tan B) ? 0. 



