If for a triangle, the radius of the circumcircle is double the radius of the inscribed circle (incircle), then which one of the following is correct?

Options

A. The triangle is a right angled.
B. The triangle is an isosceles.
C. The triangle is an equilateral.
D. None of the above

Answer

To solve the problem step by step, we will analyze the relationship between the circumradius (R) and the inradius (r) of a triangle, given that R = 2r.

### Step 1: Understanding the Definitions
- **Circumradius (R)**: The radius of the circumcircle of a triangle, which is the circle that passes through all three vertices of the triangle.
- **Inradius (r)**: The radius of the incircle of a triangle, which is the circle that is tangent to all three sides of the triangle.

### Step 2: Formulas for R and r
For any triangle:
- The circumradius \( R \) is given by the formula:
\[ R = \frac{abc}{4\Delta} \]
where \( a, b, c \) are the lengths of the sides of the triangle and \( \Delta \) is the area of the triangle.

- The inradius \( r \) is given by the formula:
\[ r = \frac{\Delta}{s} \]
where \( s \) is the semi-perimeter of the triangle, defined as \( s = \frac{a+b+c}{2} \).

### Step 3: Setting Up the Condition
According to the problem, we have:
\[ R = 2r \]
Substituting the formulas for R and r, we get:
\[ \frac{abc}{4\Delta} = 2 \cdot \frac{\Delta}{s} \]

### Step 4: Cross-Multiplying
Cross-multiplying gives us:
\[ abc \cdot s = 8\Delta^2 \]

### Step 5: Exploring the Triangle Types
To find which type of triangle satisfies this condition, we can test specific types of triangles:
1. **Equilateral Triangle**:
- For an equilateral triangle with side length \( a \):
- Area \( \Delta = \frac{\sqrt{3}}{4} a^2 \)
- Semi-perimeter \( s = \frac{3a}{2} \)
- Circumradius \( R = \frac{a}{\sqrt{3}} \)
- Inradius \( r = \frac{a}{2\sqrt{3}} \)

Substituting these values into the condition \( R = 2r \):
\[ \frac{a}{\sqrt{3}} = 2 \cdot \frac{a}{2\sqrt{3}} \Rightarrow \frac{a}{\sqrt{3}} = \frac{a}{\sqrt{3}} \text{ (True)} \]

2. **Isosceles Triangle**:
- The relationship may not hold for all isosceles triangles, as the sides can vary.

3. **Right Triangle**:
- The relationship also does not hold universally for right triangles.

### Step 6: Conclusion
From the analysis, we find that the condition \( R = 2r \) holds true for equilateral triangles. Therefore, the correct answer is:

**The triangle is equilateral.**