550 Proceedings of the Royal Irish Academy. 



Let afi meet AC m W , and ay meet AB in F, and a" be the 

 middle point of FE' ; and 0, C/ the circnmcentres of ABC, aySy. 

 1 lirough a" draw a" 0" xjarallel to AO; then 0" will be the middle 

 point of 00', and will be the centre of the Tucker's circle of ABC. 

 IsQW let-S, R' be the circumradii of the triangles ^5 C, aySy ; then we 

 have 



a"0"=^ + ^\ 



2 



Also, if pi denote the radius of Lemoine's second circle of the tri- 

 angle ABC, we have 



pi : Fa" -.-.KA: a" A : : E : ^ {E - B') ; hence 



^"' =P'[-2R 



Hence the square of the radius of the Tucker's circle of the triangle 

 AB C is equal to 



'R + R'Y ,iR- R'\- 



and it can be shown that the square of the radius of the coiTcpond- 

 ing Tucker's circle of the triangle A'B' C is the same, and they haye 

 the same centre. Hence they coincide. q.e.d. 



Peopositio:^ IX. — If the sides of the tj^iangle afSy meet the sides 

 of ABC in the points B, B' \ E, E' ; F, F', the three triangles 

 AFE', BF'B, L' CE are directly similar. Their invariaUe points 

 are the centroids of these triangles, and their double points are the other 

 points of intersection of the circle through the invaricible points with the 

 symmedians of the triangle ABC. 



Por if a, I, c be the centroids of the triangles AFE', BF'B, L' CE; 

 join aF, hF', and produce them to meet c'; and siace a is the centroid 

 of AFE', the angle KaFis the angle between two medians of AFE' ; 

 but AFE' IB similar to ABC. Hence KaF is equal to the angle 

 between tsvo medians of ABC, and therefore equal to an angle of the 

 eosymmedian triangle A'B'C, which is easily seen to be A'B'C; 

 therefore E'aF is equal to A' A C . Hence aF is parallel to A C. 

 Similarly IF' is parallel to BC. Therefore the figures KAC'B, Kac'h 

 are homothetic. Hence the point c' is on the line KC ; and it is 

 evident that the figures c'FAE', c'F'BB are directly similar. Hence 

 c' is a double point. Similarly a' , V are double points, and it is easy 

 to see that a, b, c are the invariable points. 



Cor. 1. — If we make a corresponding construction for the triangles 

 a'/S'y', A'B' C, we shall find that for the new system of three figures 

 dii'ectly similar, a, h, c are the double points ; and a', V , c' the inva- 

 riable points. Heuce the two systems are so related, that the double 

 points of either system are the invariable points of the other. 



Cor. 2. — The triangles ale, a'h'c' are cosymmedians. 



