# A circle is inscribed in a regular triangle. A square is inscribed in a circle.

**A circle is inscribed in a regular triangle. A square is inscribed in a circle. The side of the triangle is √6 larger than the side of the square. Find the side of the triangle.**

Let the length of the side of the triangle be X cm, and the side of the square is Y cm.

Then (X – Y) = √6 cm.

Y = (X – √6). (one).

The radius of a circle inscribed in a regular triangle is: R = X * √3 / 6 cm.

The radius of a circle circumscribed around a square is equal to half of its diagonal. R = Y * √2 / 2.

X * √3 / 6 = Y * √2 / 2.

Substitute Equation 1.

X * √3 / 6 = Y * √2 / 2.

X * √3 / 6 = (X – √6) * √2 / 2.

X / √3 = X / √2 – √3.

X * (1 / √2 – 1 / √3) = √3.

X * ((√3 – √2) / √6) = √3.

X = √18 / (√3 – √2) = 3 * √2 / (√3 – √2) see.

Answer: The side of the triangle is 3 * √2 / (√3 – √2) cm.