Question

The figure shows two concentric equilateral triangles with a circle within, such that the circle touches all edges of the triangle. If the radius of the circle is $\sqrt{3}$, what is the total length of the star-shaped outer border formed by the two intersecting triangles? 

Hint:

When two equilateral triangles overlap to form a star, the border length is just double the perimeter of one triangle (since both contribute equally).

Answer 1

Step 1: Understand the figure.
- A circle is inscribed inside an equilateral triangle. - Another identical equilateral triangle is rotated by 180° and superimposed, forming a star (regular hexagram).
- The border length of this star is simply the perimeter of both triangles combined.
Step 2: Relating radius of incircle to side of equilateral triangle.
For an equilateral triangle of side $a$, \[ r = \frac{\sqrt{3}}{6} \, a \] where $r$ is the inradius. Step 3: Substitute given radius.
Given $r = \sqrt{3}$, \[ \sqrt{3} = \frac{\sqrt{3}}{6} \, a \] \[ a = 6 \] Step 4: Perimeter of one triangle.
\[ P_{\triangle} = 3a = 3 \times 6 = 18 \] Step 5: Outer border of star.
Since the star is formed by two triangles, total border length = \[ 2 \times P_{\triangle} = 2 \times 18 = 36 \] Final Answer: \[ \boxed{36} \]