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The Ohio Journal of Science [Vol. XVI, No. 4, 



(27) To write such a quadrangle of h-points as noted in (26)^ 



a b c, d e f (1) 

 a c e, b d f (2) 

 a e d, c b f (3) 

 a d b, e c f (4) 



(1), (2), (3), (4) form the initial lines, properly grouped 

 for such a quadrangle. Line (2) is formed from (1) by taking 

 alternate letters, in regular order, in two groups as indicated; 

 (3) from (2) as (2) from (1); (4) from (3) as (3) from (2). The 

 h-points with (1), (2), (3), (4) as initial lines are: 



abc, def 

 \ bca, fde 

 [cab, efd 



faed, cbf 

 eda, fcb ■■ 

 [dae, bfc^ 



(hi) ; 



ace, bdf"! 

 cea, fbd 

 eac, .dfb 



(hoj. 



fadb, ecf' 

 (hg); ^dba, f ec ^ (h4). 



[bad, cfej 



Now write by 4, the g-points of the hexagons in hi, ho, hs, h^, 

 and they will be respectively: 



gi, g2, gs; 

 g3, g4, g2; 



g2, gl, gi\ 

 g4, g3, gl- 



•*• gl' g2. gs and g4 are collinear. 



(28) To write three such quadrangles of h-points (as in 27) 

 grouped about the g- point containing abcdef (1). 



Group (1) in the three ways: 



abc, def; bed, efa; cde, fab, 

 and form from each grouping a set as in (27). 



(29) The three g-points conjugate to those on the Pascals of 

 an h- point are collinear. 



They are conjugates by 5 and collinear by 19. 



(30) The three g-points of the hexagons of an h-point are 

 collinear with the g-point on the Pascal of the hexagon, uniqne to 

 the h-point. 



In 27, (h,) gave the g-points gi, ga, gs. 

 The second hexagon in h.2 gives: 



c e b f a d 



c f b d a e }- (g4) . 



c d b e a f j 



where the middle line is the hexagon unique to hi (10). 



