Hogg. — On certain Tripolar Relations. 323 



Circles of fixed radius r, whose centres are at the extremities of chords 

 of the circle ABC passing through the symmedian point of that triangle, 

 intersect on a circle whose centre is at the symmedian point, and whose 



radius is Vr 2 — E 2 tan 2 w, where w is the Brocard angle of the triangle 

 ABC. 



Consider the tw T o points P, Q, whose trilinear ratios are ( -, -, - ) and 



\A ix v) 



( — , — , — ) respectively where A + p + " = °> an ^ ^' = H-~ v > /*' = v ~ K 

 \X /x' v' / 



v = A. — fx. The two points satisfy the equation of the circle ABC — 



viz., _ + - + _ = o, and the equation of the chord PQ, AA' - + m' I- 

 a (S y a b 



+ vv - = o is satisfied by the co-ordinates of the symmedian point (a, b, c). 

 c 



Hence the tripolar equations of the two circles of radius r whose 



centres are at P, Q are 



b 2 ,„ „, c 2 



%- (X - r 2 ) + : - (Y - - r 2 ) + - (Z - - r 2 ) = o 

 A fx v 



b 2 „ 

 A' fx' 



" (X-r 2 ) + iL (Y-r 2 )+^(Z-r 2 ), 



Hence a 2 (X - r 2 ) : b 2 (Y - r 2 ) : c 2 (Z - r 2 ) 



_L 1 1 JL • JL J_ 



fxv' fx'v vA' v'A A/a' X'fj. 

 = AA : /x/x' '. w . 



Therefore 



a 2 (X -r 2 ) + 6 2 (Y - r 2 ) + c 2 (Z - r 2 ) = o 



or a 2 X +6 2 Y + c 2 Z = r 2 2(a 2 ), 



which is the equation of a circle having its centre at the symmedian point 

 of the triangle ABC. 



The equation of a circle of radius r' having its centre at the symmedian 

 point is 



K 2 (a 2 X) = k 2 6Ea6c + 2 Ar' 2 , 



where k 2 (a 2 ) = 2a 



Hence 2 (a 2 ) r 2 = K 6Ra&c + ?-^! 



48 A 2 P 2 , v / 9\ f? 

 + 2 (a 2 )r' 2 



2 o 2 i 



(a 2 ) 



Writing for 2 (a 2 ) its value 4 a cot w, where w is the Brocard angle, 

 we have 



r 2 - r' 2 = 3R 2 tan 2 oj 



i.e., r' = Vr 2 - 3R 2 tan 2 w 



It follows from the above that the length of the minimum chord 

 through the symmedian point is 2 V3 E tan w. 

 11* 



