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 ?

Question

Accepted Answer

To determine why Option 3 is the correct answer, we need to understand the relationship between the circumcircle and the incircle of a triangle, particularly in the context of an equilateral triangle.

1. **Explanation for Option 3 (The triangle is an equilateral):**

In an equilateral triangle, all sides and angles are equal. A unique property of equilateral triangles is that the radius of the circumcircle (R) is exactly twice the radius of the incircle (r). This can be mathematically expressed as:
   
   $$
   R = 2r
   $$

This relationship arises because, in an equilateral triangle, the formula for the circumradius $ R $ is given by:

$$
   R = \frac{a}{\sqrt{3}}
   $$

And the formula for the inradius $ r $ is:

$$
   r = \frac{a}{2\sqrt{3}}
   $$

Where $ a $ is the side length of the equilateral triangle. Simplifying the ratio $ \frac{R}{r} $, we get:

$$
   \frac{R}{r} = \frac{\frac{a}{\sqrt{3}}}{\frac{a}{2\sqrt{3}}} = 2
   $$

This confirms that in an equilateral triangle, the circumradius is indeed double the inradius, which matches the condition given in the question.

2. **Explanation for why the other options are incorrect:**

- **Option 1 (The triangle is a right-angled triangle):**

In a right-angled triangle, the relationship between the circumradius and the inradius is different. The circumradius $ R $ is half the hypotenuse, and the inradius $ r $ is given by the formula:

$$
     r = \frac{a + b - c}{2}
     $$

Where $ a $ and $ b $ are the legs and $ c $ is the hypotenuse. The ratio $ \frac{R}{r} $ does not simplify to 2 for all right-angled triangles, so this option is incorrect.

- **Option 2 (The triangle is an isosceles triangle):**

In an isosceles triangle, the sides and angles are not all equal, except for two sides being equal. The relationship between the circumradius and the inradius varies depending on the specific dimensions of the triangle. There is no general rule that states $ R = 2r $ for all isosceles triangles, making this option incorrect.

- **Option 4 (None of the above):**

Since we have established that the condition $ R = 2r $ holds true specifically for equilateral triangles, this option is incorrect.

In conclusion, the correct answer is Option 3, as the condition given in the question is a defining property of equilateral triangles.