HYACINTHOS 24015

[Antreas P. Hatzipolakis]:

A'b, A'c = the orthogonal projections of A on the external bisectors of B, C (ie on BIa, CIa, resp.)
 L'1 = the Euler line of AA'bA'c. Similarly L'2, L'3
 L'1, L'2, L'3 are concurrent at a point F'
 
     *** F' = X(442) complement of X(21), SCHIFFLER POINT
 Let Ia, Ib, Ic be the excenters, let Aa, Ac be the projections of A onto IaIb and IaIc, respectively, and define Bc, Ba and Ca, Cb cyclically. The Euler lines of the four triangles ABC, AAbAc, BBcBa, CCaCb concur in X(442). (Jean-Pierre Ehrmann, 11/24/01).

  *** 1.  The parallels to L'1, L'2, L'3 through  A,B,C, resp.  are concurrent  in X(79).
  
   Let Aº, Bº and Cº be the reflections of the incenter in the sides BC, CA and AB then AAº, BBº and CCº concur in X(79).. (Eric Danneels, Hyacinthos 7892, 9/13/03)
   
   
  
  *** 2.    The parallels to L'1, L'2, L'3 through  A', B', C' ,resp. (where A'B'C' = pedal triangle of I)  are concurrent  in X(3649) = KS(INTOUCH TRIANGLE), midpoint of segment X(1)X(79).
  
   Let A'B'C' be the pedal triangle of I (intouch triangle). The lines through A'.B',C' parallel to the Euler line of triangle BCI, CAI. ABI, resp. are concurrent  in X(3649)= Kirikami-Schiffler point of the triangle A'B'C'. See the note just before X(3647) in ETC.

  *** 3.    The parallels to L'1, L'2, L'3 through  Ia, Ib, Ic, resp. (where IaIbIc = excentral triangle)  are concurrent  in X(191).
  
   *** 4.  Let A''B''C'' be the extouch triangle  (where A", B", C' = the orth. proj. of Ia,Ib,Ic on BC,CA,AB,resp.).  The parallels to L'1, L'2, L'3 through  A",B",C", resp.  are concurrent  in X(3650) = KS(EXTOUCH TRIANGLE)== Kirikami-Schiffler point of the triangle  A''B''C''.
   
   Angel Montesdeoca