Problem:
The vertices of an equilateral triangle lie on the hyperbola , and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Assume without loss of generality that two of the vertices of the triangle are on the branch of the hyperbola in the first quadrant. This forces the centroid of the triangle to be the vertex of the hyperbola. Because the vertices of the triangle are equidistant from the centroid, the first-quadrant vertices must be and for some positive number . By symmetry, the third vertex must be . The distance between the vertex and the centroid is , so the altitude of the triangle must be , which makes the side length of the triangle . The required area is . The requested value is . In fact, the vertices of the equilateral triangle are , and .
The problems on this page are the property of the MAA's American Mathematics Competitions