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max(AP+BP+CP)

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  • jpehrmfr
    Dear Hyacinthists the favourite Darij s forum (mathlinks) asked for the point P on the circumcircle minimizing AP + BP + CP. It is quite easy to see that this
    Message 1 of 2 , May 31 2:45 PM
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      Dear Hyacinthists
      the favourite Darij's forum (mathlinks) asked for the point P on the
      circumcircle minimizing AP + BP + CP. It is quite easy to see that
      this point is a vertex of ABC.
      More interesting is the point P of the circumcircle maximizing AP + BP
      + CP.
      If A'B'C' is the medial triangle and M a point of the arc B'C' (not
      containing A') of the NP circle, an easy barycentric computation gives
      a funny result :
      the power of M wrt the A-excircle is (A'M + B'M + C'M)^2 thus A'M +
      B'M + C'M can be extremal only at B', C' or at the reflection of Fa in
      the NP-center (Fa = contact point of the NP circle with the A-excircle)
      Using the homothecy (G,-2), it follows immediately that the required
      point P is the reflection of H in Fm, where Fm is the contact point of
      the NP circle with the greatest excircle.
      More over Max(AP+BP+CP) = 2 OJ = 2 root(R(R+2R')) where J and R' are
      the center and the radius of the greatest excircle.
      Friendly. Jean-Pierre
    • Antreas P. Hatzipolakis
      ... Dear Jean-Pierre Some quick generalizations: 1. min/max for P lying on a well-defined circumellipse. 2. ABCD := a quadrilateral inscribed in a circle.
      Message 2 of 2 , Jun 4, 2005
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        [JPE]:
        >the favourite Darij's forum (mathlinks) asked for the point P on the
        >circumcircle minimizing AP + BP + CP. It is quite easy to see that
        >this point is a vertex of ABC.
        >More interesting is the point P of the circumcircle maximizing AP + BP
        >+ CP.
        >If A'B'C' is the medial triangle and M a point of the arc B'C' (not
        >containing A') of the NP circle, an easy barycentric computation gives
        >a funny result :
        >the power of M wrt the A-excircle is (A'M + B'M + C'M)^2 thus A'M +
        >B'M + C'M can be extremal only at B', C' or at the reflection of Fa in
        >the NP-center (Fa = contact point of the NP circle with the A-excircle)
        >Using the homothecy (G,-2), it follows immediately that the required
        >point P is the reflection of H in Fm, where Fm is the contact point of
        >the NP circle with the greatest excircle.
        >More over Max(AP+BP+CP) = 2 OJ = 2 root(R(R+2R')) where J and R' are
        >the center and the radius of the greatest excircle.

        Dear Jean-Pierre

        Some quick generalizations:

        1. min/max for P lying on a well-defined circumellipse.

        2. ABCD := a quadrilateral inscribed in a circle.
        min/max of AP + BP + CP + DP (for P on the circle)

        3. ABCD := a tetrahedron min/max of AP + BP + CP + DP
        for P on the circumsphere of ABCD.


        APH
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