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This is some very hard geometry

 

Triangle XYZ is equilateral.  Points Y and Z lie on a circle centered at O, such that X is the circumcenter of triangle OYZ, and X lies inside triangle OYZ.  If the area of the circle is , then find the area of triangle XYZ.

 May 21, 2023
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Let's call the radius of the circle R. Since X is the circumcenter of triangle OYZ, it lies on the perpendicular bisectors of all three sides of the triangle. This means that XY = XZ = R, and YZ = 2R/sqrt(3), which is the diameter of the circle.

Now let's find the area of triangle XYZ. We can use the formula for the area of an equilateral triangle:

Area of XYZ = (sqrt(3)/4) * XY^2

Substituting XY = R, we get:

Area of XYZ = (sqrt(3)/4) * R^2

To find the area of the circle, we use the formula:

Area of circle = pi * R^2

Substituting the given area, we get:

pi * R^2 = 

Solving for R, we get:

R = sqrt() / pi

Substituting this value of R into the formula for the area of XYZ, we get:

Area of XYZ = (sqrt(3)/4) * (sqrt() / pi)^2
Area of XYZ = (3 / 4pi) * 

Simplifying, we get:

Area of XYZ = (3sqrt(3)) / 4

Therefore, the area of triangle XYZ is (3sqrt(3)) / 4. 

 May 22, 2023

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