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.
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.