Casey — 0)i the Harnioinc Hexagon of a Triangle. 547 



Let O, fl' be the Brocard points of ABC; then the angle QOK= o). 

 = 0)'. Hence (2 is a Brocard point of the triangle A'B' C. 



Cor. 4. — If X, ^, z; x', tj', z' be the perpendiculars let fall from 

 their common symmedian point on the sides of the two cosymmedian 

 triangles ABC, A'B'C; 



., X x' 11 y' s z' 



then - = _ = ^ = I = - = 



a a b c c 



For - = 2 tan w, —,= 2 tan w'. Hence - - — . 



a a a a 



Proposition IV. — If the circumcircle of two cosymmedian triangles 

 be inverted from any arbitrary point, the inverses of their six vertices 

 will be the vertices of tivo other cosymmedian triangles. 



Bern. — Let the points inverse to A, B, C; A' , B' , C, respectively, 

 be ^1, Bi, Ci ; Ai, B^', C/. Now since the anharmonic ratio of any 

 four concyclic points is equal to the anharmonic ratio of the four 

 points inverse to them (Sequel, Book VI., Sect, iv., Prop. 8) ; 

 the four points Bi, Ai, Ci, Ai form a harmonic system. Hence 

 AiAi is a symmedian of the triangle AiBiCi, and the proposition 

 is proved. 



Cor. 1. — The circumcircle of two cosymmedian triangles can be 

 inverted, so that the points inverse to their vertices will be the 

 vertices of two equilateral triangles. 



For if the inverses oi A, B, C be the vertices of an equilateral 

 triangle, the inverses of A', B', C will evidently be the vertices of 

 another ; but if the centre of inversion be either of the two points 

 common to two of the Apollonian circles of the triangle ABC, it will 

 invert into an equilateral triangle. Hence the proposition is proved. 



Cor. 2. — "When two cosymmedian triangles invert into two 

 equilateral triangles, the harmonic hexagon inverts into a regular 

 hexagon. 



Cor. 3. — The inverse of the angular points of a harmonic hexagon 

 from any arbitrary poiat are the angular points of another harmonic 

 hexagon. 



Pkoposition V. — The anharmonic ratio of any four consecutive ver- 

 tices of a harmonic hexagon is constant. 



Bern. — The anharmonic ratio is equal to that of four consecutive 

 vertices of a regular hexagon, which is constant. 



Cor. 1 . — Any side of a triangle is divided in a given anharmonic 

 ratio by the two non-corresponding sides of its cosymmedian. 



For consider the line AB ; the anharmonic ratio of the four points 

 ALMB is equal to the pencil ( C. AB'A'B), which is given, being 

 equal to that of a corresponding pencil for a regular hexagon. 



