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Re:Quadrilateral problem, Re: Quadrilateral problem: PROOFS

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  • rafinad2003
    Dear Darij, ... My first question in previous message was about adjusting original ABCD to make it homothetic to Alexey s A B C D . This adjustment will
    Message 1 of 41 , Jan 15, 2004
      Dear Darij,

      >
      > Alas, I don't understand 1. and 2., and have no
      > answers to 3. and 4.; however, the analogy with
      > centroids would be quite unusual. I am still
      > hoping to find a nice expression for the radius
      > of (O'), but I don't see any direct way to this.
      >

      My first question in previous message was about 'adjusting'
      original ABCD to make it homothetic to Alexey's A"B"C"D".
      This 'adjustment' will move the center of the original incircle
      onto line O'P. Will it change it's radius also?
      Now, along with incenters X,Y,Z,W we also looked at the
      ex-centers Xe,Ye,Ze,We. Quadrilateral XeYeZeWe is homothetic to
      XYZW (?). When we 'adjust' side AB to A1B1, A1B1 will be tangent
      still to circle X and the new incircle of A1B1C1D1.
      The new side A1B1 will not be tangent to ex-circle as old AB was.
      The configuration of ex-circles has to 'adjust' all the way up.
      But if XeYeZeWe was homothetic to XYZW before, it will not
      be after this 'ripple'. Will it?


      The second one was about the possible ex-circle that touches
      the continuations of all 4 sides of ABCD . The 'adjusted' A1B1C1D1
      being bicentric, maybe has a better chance of having
      this ex-circle.


      Sincerely,
      Rafi.
    • rafinad2003
      Dear Darij, ... My first question in previous message was about adjusting original ABCD to make it homothetic to Alexey s A B C D . This adjustment will
      Message 41 of 41 , Jan 15, 2004
        Dear Darij,

        >
        > Alas, I don't understand 1. and 2., and have no
        > answers to 3. and 4.; however, the analogy with
        > centroids would be quite unusual. I am still
        > hoping to find a nice expression for the radius
        > of (O'), but I don't see any direct way to this.
        >

        My first question in previous message was about 'adjusting'
        original ABCD to make it homothetic to Alexey's A"B"C"D".
        This 'adjustment' will move the center of the original incircle
        onto line O'P. Will it change it's radius also?
        Now, along with incenters X,Y,Z,W we also looked at the
        ex-centers Xe,Ye,Ze,We. Quadrilateral XeYeZeWe is homothetic to
        XYZW (?). When we 'adjust' side AB to A1B1, A1B1 will be tangent
        still to circle X and the new incircle of A1B1C1D1.
        The new side A1B1 will not be tangent to ex-circle as old AB was.
        The configuration of ex-circles has to 'adjust' all the way up.
        But if XeYeZeWe was homothetic to XYZW before, it will not
        be after this 'ripple'. Will it?


        The second one was about the possible ex-circle that touches
        the continuations of all 4 sides of ABCD . The 'adjusted' A1B1C1D1
        being bicentric, maybe has a better chance of having
        this ex-circle.


        Sincerely,
        Rafi.
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