Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.8 AIME Find the answer

Unit circle $\Omega$ has points $X, Y, Z$ on its circumference so that $X Y Z$ is an equilateral triangle. Let $W$ be a point other than $X$ in the plane such that triangle $W Y Z$ is also equilateral. Determine the area of the region inside triangle $W Y Z$ that lies outside circle $\Omega$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Let $O$ be the center of the circle. Then, we note that since $\angle W Y Z=60^{\circ}=\angle Y X Z$ , that $Y W$ is tangent to $\Omega$ . Similarly, $W Z$ is tangent to $\Omega$ . Now, we note that the circular segment corresponding to $Y Z$ is equal to $\frac{1}{3}$ the area of $\Omega$ less the area of triangle $O Y Z$ . Hence, our total area is $\frac{3 \sqrt{3}-\pi}{3}$

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