Problem:
An equilateral triangle is inscribed in the ellipse whose equationis . One vertex of the triangle is , one altitude is contained in the -axis, and the length of each side is , where and are relatively prime positive integers. Find .
Solution:
Let the other two vertices of the triangle be and , with . Then the line through and forms a -degree angle with the positive -axis, and its slope is . Therefore, the line's equation is . Substituting this into the equation of the ellipse and simplifying yields
The triangle has sides of length , and .
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Let the other two vertices of the triangle be and , with . Equating the squares of the distances from to and from to yields
Substituting from the equation of the ellipse, it follows that . The roots of this quadratic are and . If , then , so . Solving for yields , so that the triangle has sides of length , and .
Query: There are two other equilateral triangles with one vertex at that are inscribed in the ellipse . Can you find the lengths of their sides?
The problems on this page are the property of the MAA's American Mathematics Competitions