NOTE ON TRILINEARS. 249 



NOTE ON TRILINEARS. 



1. The property of transversals. 



With the usual notation, let the sides c, a, b of the triangle of re- 

 ference ABC, taken in order, be divided by the points F, D, E, 

 respectively in the ratios p : q, g^ : r, r : p» Then the equations to the 

 lines CF, AD, BE respectively are 



p a a — q b (3 =: 0, 

 q b (3 — r c y =0, 

 r c y — p a a = 0, 



and the three lines meet in the point 



p a a = q b j3 = r c y. 



Again, the equations to the lines DE, EF, FD are 



paa + ff^/S — r c y = 0, 

 q b (3 + r c y — p a a = 0, 

 r c y + p aa — q b (3 ^ 0, 



and if these lines be produced to meet each the remaining side of the 

 triangle, the three points of section lie in the line 

 paa-\-qbft-]-rcy = 0. 



Also, the equations to the lines joining these points of section each 

 to the remaining vertex of the triangle, are 



paa-{-qb^ = 0, 



qbp-\-rcy = 0, 



r c y -\- p a a = 0, 

 and these with the sides of the triangle and the three first-mentioned 

 lines form harmonic pencils at the vertexes of the triangle. 



Cor. 1. This includes the following well-known cases : 



(1) If p=l, 2=1, r=l, we have the bisecters of the sides. 



(2) If p=b cos A, &c, we have the perpendiculars from the 



angles on the sides. 



(3) If p=b, q=c, r=a, we have the bisecters of the angles. 



(4) If p=cot -X , &c,, we have the lines from the angles to the 



points of contact of the inscribed fircle. 

 Vol. IX. R 



