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Fwd: Re: two excircles

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  • Andreas P. Hatzipolakis
    ... -- [Non-text portions of this message have been removed]
    Message 1 of 1 , Jul 2, 2007
      >Date: Mon, 02 Jul 2007 08:33:21 -0000
      >From: "alexey_zaslavsky" <alexey_zaslavsky@...>
      >To: Hyacinthos-owner@yahoogroups.com
      >Subject: Re: two excircles
      >
      >Dear Jean-Pierre and Francois!
      >>
      >>
      >> Here are some remarks (I've changed the notations and I didn't find
      >a
      >> synthetic proof)
      >> Let I,Ia,Ib,Ic be the incenter and excenters; U,U' the feet on BC
      >of
      >> the internal and external A-bisectors
      >> Then there exists
      >> - an hyperbola h with center U going through the 4 projections of I
      >and
      >> Ia upon AB and AC (the asymptots are BC and the other common
      >tangent
      >> through U to the incircle and the A-excircle)
      >> - an hyperbola h' with center U' going through the 4 projections of
      >Ib
      >> and Ic upon AB and AC (the asymptots are BC and the other common
      >> tangent through U' to the B- and C-excircles)
      >>
      >> Consider a point M lying on h; the parallel and antiparallel to BC
      >> through M intersect AB and AC at four points lying on a circle, and
      >> this circle touches the incircle and the A-excircle (and the line
      >> through the contact points goes through U)
      >> Consider a point M' lying on h'; the parallel and antiparallel to
      >BC
      >> through M' intersect AB and AC at four points lying on a circle,
      >and
      >> this circle touches the B-excircle and the C-excircle (and the line
      >> through the contact points goes through U')
      >>
      >I can propose a simple construction of circles touching A-excircle
      >and B-excircle of the triangle ABC.
      >Let P be an arbitrary point on the radical axis of excircles, C_0 ---
      >the midpoint of AB. Q is the common point of perpendicular from P to
      >AB and CC_0. The line passing through Q and parallel to AB intersect
      >AC and BC in points X and Y. Then the circle XPY touches two
      >excircles.
      >In fact as P in the radical axis there exists an inversion with
      >center P conserving both excircles. It isn't difficult to note that
      >this inversion transforms the circle XPY to the line AB.
      >
      >Sincerely Alexey


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