670 Tra/is(tcfio/is. 



The co-ordinates of the points X', Y', Z' are reBpectiveiy 



(o. A|, Af). (a;;, „, ^. (a?, a|, „) 



where A. = -- — , A being the area of the triangle of reference. 



j> + q » 



The co-ordinates of the points A', B', C are respectively 



(o,/4,/^f)-(^f.4o), (^?,4o) 



where jj. 



P- 1 



It follows that the co-ordinates of the centroids of the triangles 



X'Y'Z', A'B'C are { ^, ^, -7;— |, which are those of the centroid of the 

 ' \ 3a 3o 3c / 



triangle ABC. Since this result is independent of ^j and q we see that 



all triangles formed in this manner are co-centroidai with the triangle 



ABC. 



Let now the sides Y'Z', Z'X', X'Y' be divided in X^, Y^, Z^ so that 



Y'X,: X,Z' = Z'Y, : Y,X' = X'Z, : Z,Y' = v - 1, 



then the co-ordinates of the points X^, Y^, Z^ are respectively 



- X -2 A 



where v 



P + Q {P + '!}- 



Hence the triangle X^Y^Z^ is also co-centroidal with the triangle ABC. 

 This result also holds for a triangle similarly formed by dividing the 

 sides of the triangle A'B'C, and the process may evidently be continued 

 indefinitely. 



2. The following simple relations may be easily proved : — 



If A, A', A, be the areas of the triangles ABCj X'Y'Z', A'B'C 

 respectively, then 



{p + g.)^ 



^ p^+pq + q"^ 

 (p-q)^ 



if A' = iiA, the minimum value of n is 7, in which case p = q, and 

 the triangle X'Y'Z' is the medial triangle of the triangle x\BC. 



The triangles AY'Z', BZ'X', CX'Y' are equal in area, the common 



. vq 



value bemg j^-jy, A . 



The triangles AB'C, BC'A', CA'B' are equal in area, the common 

 , . pq 



value bemg 7-^^-T^ A . 



The centroids of the triangles AY'Z', AB'C, as the ratio p : q 

 varies, lie on straight lines parallel to BC and bisecting AG, w^here G is 

 the centroid of the triangle ABC. 



The middle points of the sides of all centroidal triangles lie on the 

 ■sides of the menial triangle of the triangle ABC. 



