554 Pi-oeeedinys of the Roijal Irhh Academy. 



Bern. — 2 tan © = LL' ^ A'B = perpendicular from K on A'B ^ A'B; 

 and 2 tan w = perpendicular from K on BC ^ BC . Hence 



tan _ peip. from K on A'B B C 

 tan w perp. fi'om ^on BC A'B 



smA'BX BC A'B' BC 



smCBK A'B B'C ^ A'B' 



that is, equal to the anharmonic ratio of the four points B, A', C, B', 

 and therefore equal to the anharmonic ratio of the corresponding 

 points in a regular hexagon, and therefore equal to 3. Hence 

 tan ® = 3 tan co. 



Peopositioij XIII. — i/" figures directly similar le described on the 

 Mdes of the harmonic hexagon, the middle points of its symmedian lines 

 AA', BB', CC are each a double point for three pairs of figures. 



Bern. — Let A" he the middle point of AA'-, then it may he proved, 

 as in Conies, page 247, that A" is the douhle point of the figures on 

 C'A, AB'; and also of the figures described on BA', A' C ; and it 

 remains to he proved that it is a douhle point of the figures described 

 €n BC, B'C. Join BA/' A"C; C'A", A"B; then, since A" is a 

 double point of the figures C'A, AB', we have C'A" : A"A : : A" A 

 : A"B'. Hence C'A" . A"B' = A"A-. Similarlr BA" . A" C = A" A'-. 

 Hence BA" . A"C= C'A" . A"B'; and the angles BA"C', B'A"C are 

 equal. Hence the triangles BA"C', B'A"C are equiangular, and 

 they are directly similar. Hence the proposition is proved. 



Cor. 1. — The four points B, C, K, A" are concyclic. 



Cor. 2. — If figures directly similar be described on the six sides of 

 the hannonic hexagon AB'CA'BC, the symmedian lines of the 

 harmonic hexagon, formed by any six corresponding lines, pass 

 respectively through the middle points A!' , B", C" of the symmedian 

 lines of ^^'C^'^C. 



Cor. 3. — In the same case, the locus of the symmedian point of 

 the hexagon, formed by six corresponding lines of these figures, 

 is the Brocard circle of the original hexagon. 



Cor. 4. — The centre of similitude of any two hexagons, each 

 formed by six corresponding lines of figures directly similar described 

 on the sides of AB' CA'BC, is a point on its Brocard circle. 



Cors. 2, 3, 4 may be proved exactly as in the coiTesponding cases 

 for triangles (see Conies, page 248). 



Cor. 5 — The six lines joining, respectively, the invariable points 

 L, If, N, P, Q, it to six corresponding points are concun-ent. Tlie 

 locus of theii' point of concurrence is the Brocard circle of AB' CA'BC, 

 and they fonn a pencil in involution. 



Proposition XIY. — TJw triangle formed by three alternate sides of 

 the harmonic hexagon is in perspective with the triangle formed by the 

 three invarialle points corresponding to the three remaining sides. 



