1976 AMC 12 Problem 22

Question

Given an equilateral triangle with side of length

s,

consider the locus of all points

P

in the plane of the triangle such that the sum of the squares of the distances from

P

to the vertices of the triangle is a fixed number

a.

This locus

is a circle if

a\gt s^2

contains only three points if

a=2s^2

and is a circle if

a\gt2s^2

is a circle with positive radius only if

s^2\lt a\lt2s^2

contains only a finite number of points for any value of

a

is none of these

Let

G

be the centroid and express the sum using

PG^2

For an equilateral triangle, the sum equals

3PG^2+s^2

For vertices

A,

B,

and

C

with centroid

G,

put

T=GA^2+GB^2+GC^2.

The standard centroid identity gives

PA^2+PB^2+PC^2=3PG^2+T.

In an equilateral triangle, each

GA=\frac{s}{\sqrt3},

so

T=s^2.

Hence

3PG^2=a-s^2.

The locus is empty for

a\lt s^2,

one point for

a=s^2,

and a circle of positive radius for

a\gt s^2.

Therefore, the correct answer is A.

Problem 22 in other years