Feuerbach hyperbola

      > 

      [AH?]> Consider the center O instead of I, and obtain the hyperbola as a locus.

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      >                    Theorem of Neuberg - Mandart  (1893)

      [JHC]   I'll translate as I go:

      > Sur les  mediatrices A'O, B'O, C'O du triangle ABC, 

      on the  edge-bisectors? (in which case A'B'C' will be the subtriangle)

      > on prend trois longueurs egales: > 

      one takes three equal lengths

      >                     A'P_a = B'P_b = C'Pc = l,

      > 

      > toutes trois vers O, ou dans le sens oppose, et l'on trace le triangle

      all towards  O  or all away from  O,     and draws the triangle

      > P_aP_bP_c; les perpendiculaires abaissees de A, B, C, sur les cotes

      then  the perpendiclars from  A,B,C  to the sides of Pa Pb Pc

      se coupent en un meme point P,

      will meet in a point  P

      > dont le lieu, lorsque l varie, est l'hyperbole de Feuerbach.

      whose locus is the Feuerbach hyperbola.

      Aha!  I remark that one can get the other Feuerbach (or Boutin) hyperbolae by taking some of them in one sense, some in another.     

      I'll have to check on the validity of my translation (of "mediatrices") by verifying this theorem.  

      The naming problem for these Feuerbach or Boutin hyperbolae will have to be faced.  I rather prefer "Feuerbach", but it's really wrong, because of course Feuerbach didn't discuss them, and Boutin did.  Perhaps the cure is to call them Feuerbach-Boutin hyperbolae and have a piece of text saying why.  But I really prefer to use meaningful adjectives rather than these unhelpful personal names.

      Here's a possible idea - what do you think?  We introduce the term "Pickwickian hyperbola" for the conic through  A,B,C,H  and the Pickwickian point  P.  Then these would be (Boutin's) "Incentral (and excentral) hyperbolae"    ABCH Ix.  There'd also be

      Jerabek's "circumcentral hyperbola"    ABCHO

      and

      Kiepert's "centroidal hyperbola"     ABCGH