674 Transactions. 



It is seen by inspection that if K be the svmmedian point of the 

 triangle ABC, then H', H", H'" He on AK, BK, CK respectively. 



The trilinear ratios of the middle point of the chord of the circle 

 ABC drawn from A through K are 



(26c cos A 7 \ 



Hence we see that the three foci H', H'', H'" lie on the Brocard circle 

 or circle having as its diameter the Ime joining the centre of the circle 

 ABC to the symmedian point K. 



If F be the centre of the circle ABC, then H' lies on the circle whose 

 diameter is AF — viz., the circle 



{c/3 + by)'L-2aS = 0. 



It may also be shown that the four points BCFH' are concyclic. 



If the tangent to the circle ABC at A meets BC in T, then H' is the 

 foot of the perpendicular from A on FT. 



The equation of the Brocard circle may be written 



'^bc(l + l + l)L-i{a').S = o. 



and it is easily seen that it is satisfied by the co-ordinates of H', H", 

 and H'". 



7. If 8j, 8.2, 83 be the medians drawn from A, B, C respectively to the 

 middle points of the opposite sides of the triangle ABC, then the semi- 

 latera recta of S', S", S'" are respectively 



A« a; a_2 



5i3' 5.2«' 538' 

 where A is the area of the triangle ABC. 



Also, AH' = |, BH' = |, CH' = |, 



whence AH'^ = BH'. CH'. 



We have AK = J^ . 28^, 



and therefore AH' . AK = ^,— „,. 



l(a2) 



Hence if T', T", T"' be the lengths of the tangents from A, B, C 

 respectively to the Brocard circle, 



abc 



aT = hT" = cV" = —7==r. 



The directrices of the parabolas S', S", S'" are respectively 

 Dj = a cos Aa — 0/3 — by — 

 Do = — ca + /> cos B/3 — ay ■— 

 Dy = — ba — afS + c cos Cy = 0. 

 From the form of D^ it is seen that it passes through the point in which 

 the tangent at A to the circle ABC meets BC. 



Let the vertices of the triangle formed by Dj, D2, D3 be Vi,V2,Vs. 

 The equations of the lines AV^, BV.^, CV., are respectively 

 P {ca + b^ cos B) - y {ab + c^ cos C) = 

 y (ab + c" cos C) — a {be + a^ cos A) = 

 a (be + a* cos A) — /B{ca + &=• cos B) = 0. 



