A circle is inscribed in an equilateral triangle with side lengths 6 unit. Another circle is drawn i

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A circle is inscribed in an equilateral triangle with side lengths 6 unit. Another circle is drawn inside the triangle (but outside the first circle), tangent to the first circle and two of the sides of the triangle. The radius of the smaller circle is:

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Sol. [Key Concept: Incircle of an equilateral triangle has $r = \frac{R}{2}$; chain the radii for the inscribed sequence.]$$	ext{~}	ext{now}	ext{~}\begin{matrix} & x = \sqrt{3} - r = PB \ & \frac{\sqrt{3} - r}{\sqrt{3} + r} = sin30^{\circ} = \frac{1}{2} \ & \frac{r}{\sqrt{3}} = \frac{2 - 1}{2 + 1} = \frac{1}{3} \ & r = \frac{1}{\sqrt{3}} \end{matrix}$$

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