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Re: [EMHL] Altitudes as diameters of circles [variation]

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  • Antreas P. Hatzipolakis
    On Thursday, February 7, 2002, at 02:00 PM, Antreas P. Hatzipolakis ... BHb, CHc ... Similarly define the circles (Kbc),(Kba);(Kca),(Kcb) H a = the 2nd
    Message 1 of 20 , Feb 7 3:48 PM
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      On Thursday, February 7, 2002, at 02:00 PM, Antreas P. Hatzipolakis
      wrote:

      > Another variation:
      >
      > AHa, AHb, AHc = altitudes.
      BHb, CHc
      >
      > Hab = Orth. Proj. of Ha on AB,
      > Hac = Orth. Proj. of Ha on AC.
      >
      >
      > (Kab) = the circle with diameter HaHab
      > (Kac) = the circle with diameter HaHac

      Similarly define the circles (Kbc),(Kba);(Kca),(Kcb)

      H'a = the 2nd intersection of (Kab),(Kac)
      [the first is Ha]
      that is, HaH'a is the common chord of (Kab),(Kac)

      H'b = the 2nd intersection of (Kbc),(Kba)
      [the first is Hb]

      H'c = the 2nd intersection of (Kca),(Kcb)
      [the first is Hc]

      Are the triangles

      1. HaHbHc, H'aH'bH'c

      2. ABC, H'aH'bH'c

      perspective?


      Antreas


      >
      > A'= (2nd tangent to (Kab) from B) /\ (2nd tangent to (Kac) from C)
      > [the 1st tangent is AB] [the 1st tangent is AC]
      >
      > Similarly B', C'.
      >
      > A
      > /\
      > / \
      > / \
      > / \
      > / \
      > / \
      > / \
      > Hab \
      > / Hac
      > / Kab \
      > / Kac \
      > B-----------Ha----------C
      >
      > A'
      >
      >
      >
      > The triangles ABC, A'B'C' are perspective.
      >
      > 1
      > Perspector: ( ----------------- ::) = (tanAsin^2A ::) in Barycentrics.
      > cotA*(1 + cot^2A)
      >
      >
      > Antreas
      >
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