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?

For an equilateral triangle with side length 6 units, the altitude is 6 * sqrt(3) / 2 = 3 * sqrt(3), and the inradius of the circumscribed circle is one-third of the altitude, making it sqrt(3). The smaller circle sits in the corner of the triangle and is tangent to two sides and the larger circle. By creating a smaller similar equilateral triangle at the vertex and using the geometric properties of tangent circles (Soddy circle formula for two lines and a circle), the radius of the smaller circle R is given by the formula R = r / (2 * sqrt(3) + 1), which evaluates to 1/sqrt(3).

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