- 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 - [JPE]:
>the favourite Darij's forum (mathlinks) asked for the point P on the

Dear Jean-Pierre

>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.

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

--