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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.

 Apr 30, 2023
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Lets O be the center of the circle. From point O, draw a line perpendicular to XY such that it will intersect the circle at Z. That will be the greatest possible perimeter of triangle XYZ. 

Area of the circle = (pi)r^2 = 18pi

so, r = 3sqrt2 = OX = OY = OZ (all are radius of the circle)
YZ^2 = OY^2 + OZ^2 = (3sqrt)^2 + (3sqrt)^2 = 36
YZ = 6

diameter, XY= 6sqrt2
XZ = YZ = 6
so perimeter = 6 + 6 + 6sqrt2 = 12 + 6sqrt2. 

 May 1, 2023

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