Area of the largest square inscribed in an equilateral triangle that is itself inscribed in a circle of radius $r$

Question

$\triangle ABC$ is an equilateral triangle inscribed in a circle of radius $r$. What is the area of the largest square that can be inscribed inside it?

My doubt: How side of an equilateral triangle will be $r\sqrt{ 3}$

Answer 1

Hint: Draw the radii from the center of the circle to the vertices of the triangle. Do you notice anything special about the resulting three smaller triangles? What are their angles?

Answer 2

$\angle OBC = 30^\circ$, Hence $BM=r\cos 30^\circ = \frac{\sqrt3 r}{2}.$

Task given to you, find the length of $BC$.