## max(AP+BP+CP)

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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
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
• ... 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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