MathDB

In space, we are given the points

A,B,C

and a sphere with center

O

and radius

1

. Find the point

X

from the sphere for which the sum

f(X)=|XA|^2+|XB|^2+|XC|^2

attains its maximal and minimal value. Prove that if the segments

OA,OB,OC

are pairwise perpendicular and

d

is the distance from the center

O

to the centroid of the triangle

ABC

then:
(a) the maximum of

f(X)

is equal to

9d^2+3+6d

; (b) the minimum of

f(X)

is equal to

9d^2+3-6d

.
K. Dochev and I. Dimovski

Problem Statement