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$\triangle ABC$ 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 $r\sqrt{ 3}$

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  • $\begingroup$ Do you know how to solve further if you get $a=r\sqrt 3$? $\endgroup$ Commented Aug 1, 2017 at 20:59
  • $\begingroup$ Yes @JaideepKhare $\endgroup$
    – EmilySekuz
    Commented Aug 1, 2017 at 21:04

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

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

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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?

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  • $\begingroup$ Their angle will be 30 degree @Yushwuth $\endgroup$
    – EmilySekuz
    Commented Aug 1, 2017 at 21:05
  • $\begingroup$ So, do you know anything special about 30-60-90 triangles? (use trigonometry) $\endgroup$
    – Legendre
    Commented Aug 1, 2017 at 21:06

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