1962 AMC 12 Problem 15

Question

Given triangle

ABC

with base

AB

fixed in length and position. As the vertex

C

moves on a straight line, the intersection point of the three medians moves on:

a circle

a parabola

an ellipse

a straight line

a curve here not listed

The centroid lies two-thirds of the way from a vertex to the midpoint of the opposite side

With

AB

fixed, express the centroid as a fixed point plus one-third of the position vector of

C

Let

A,

B,

and

C

also denote their position vectors. The centroid is

G=\frac{A+B+C}{3}.

Since

A

and

B

are fixed, this is a translation and scaling by

\frac{1}{3}

of the position of

C.

A straight-line locus therefore maps to a straight-line locus.

Thus, the correct answer is D.

Problem 15 in other years