1. **Stating the problem:** We have a triangle ABC with an inscribed circle and an internal equilateral triangle with side length 2 cm. The triangle is extended to form a larger triangle ABD where sides AD and BD are each 8 cm. We want to analyze the geometric relationships and possibly find unknown lengths or angles. 2. **Understanding the given data:** - Side of the internal equilateral triangle: $2$ cm - Sides $AD = 8$ cm and $BD = 8$ cm 3. **Analyzing the triangle ABD:** Since $AD = BD = 8$ cm, triangle ABD is isosceles with $AD = BD$. 4. **Using the properties of the equilateral triangle inside ABC:** Each side of the equilateral triangle is $2$ cm, so all its angles are $60^\circ$. 5. **Position hint 'center' suggests the equilateral triangle is centered inside the circle inscribed in ABC.** 6. **If needed, use the Law of Cosines or other geometric relations to find unknown lengths or angles depending on the specific question.** Since the problem does not specify a particular unknown to solve, this is the analysis based on the given data.