€72 Transactions. 



The lines L', Li intersect on BC ; M', M] intersect on CA ; and N', Nj 

 intersect on AB. Calling these points of intersection A", B", C" respec- 

 tively, we have for the equations of the lines B"C", C"A", A"B" 



L" = p-qhia + p^b^ + q^cy — o 

 M" = q^aa + if(fbji + p'cy = o 

 N" — p^aa + q^b^ + p'q-^^^y = 0. 



Comparing the equations L", M", N" with those of L', M', N' we see 

 that the triangle A"B"C" is a centroiclal triangle formed by dividing the 

 sides of the triangle ABC in the i-atio q^ -.p^. 



The area of the triangle A"B"C" is given by 



and therefore the areas of the triangles A"B"C", A'B'C, X'Y'Z', and 

 ABC are connected by the relation 



A " . A =^ A ' . A , . 



If BC, CA, AB be divided internally in X", Y", Z" so that (A"BX"C), 

 (B"CY"A), (C"AZ"B) form harmonic ranges, we have a fourth centroidal 

 triangle X"Y"Z", inscribed in the triangle ABC, the equations of whose 

 sides, Lo, M2, No, may be formed from L", M", N" l)y writing — g^ for q^ in 

 the latter equations. 



4. Let P'Q'K', P"Q"R", PiQiRj, P,Q,'R-. be respectively the poles of 

 L'M'N', L"M"N", L^M^Ni, L.^MaN., with regard to the triangles. 



The co-ordinates of P'Q'R' are proportional to 



(— , -- --^ (-^ — , -^) (--, ~ —) 



These points are the vertices of the triangle formed by the lines 

 AX', BY', CZ' ; as the ratio }) : q varies the loci of these points are the 

 •ellipses 



Sj = a^o~ — bc-Py = 



52 = b^l^^ — cay a = 



53 = c%^ — abaji = 0. 



The lines AP', BQ', CR' will meet S^, S.,, S, respectively in P^, Q^, R,. 

 The position of P^ may be found by observing that (B . AP'CPj^) is an 

 harmonic pencil. Q^ and R^ may be found in a similar manner. 



The lines BB', CC meet in P,; CC, AA' in Q., ; AA', BB' in R,. 

 P", Q", R" may be found from P.^, Q.^, Ro in the manner employed to deter- 

 mine PiQiRj. 



The four triangles P'Q'R', P"Q"R", PiQiRi, and P,Q,R-2 have their 

 •centroids at the point G. 



5. The lines L', L^ may be respectively written 



L' = pq (aa -f 6/3 + Cy) — {i) - q){qb/3 — pcy) — 

 Li = - pq (aa + b(3 + Cy) + {p + q)iqb(5 + pcy) = 0. 

 The equations of AX', AA' are respectively 



pbfS — qcy = 



pbfS + qCy — 0. 



