Problem:
Six points are equally spaced around a circle of radius . Three of these points are the vertices of a triangle that is neither equilateral nor isosceles. What is the area of this triangle?
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Solution:
The six points divide the circle into six arcs each measuring . By the Inscribed Angle Theorem, the angles of the triangle can only be , and . Because the angles of the triangle are pairwise distinct the triangle must be a triangle. Therefore the hypotenuse of the triangle is the diameter of the circle, and the legs have lengths and . The area of the triangle is .
The problems on this page are the property of the MAA's American Mathematics Competitions