If the triangle be equilateral, then `R:r:r_1 =`

Options

A. `1:1:1`
B. `2:1:3`
C. `1:2:3`
D. `3:2:4`

Answer

To solve the problem of finding the ratio \( R : r : r_1 \) for an equilateral triangle, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Variables**:
- \( R \): Circumradius of the triangle.
- \( r \): Inradius of the triangle.
- \( r_1 \): Exradius of the triangle.

2. **Finding the Circumradius \( R \)**:
- For an equilateral triangle with side length \( a \):
\[ R = \frac{abc}{4\Delta} \]
Since \( a = b = c = a \) for an equilateral triangle, we can write:
\[ R = \frac{a^3}{4\Delta} \]
- The area \( \Delta \) of an equilateral triangle is given by:
\[ \Delta = \frac{\sqrt{3}}{4} a^2 \]
- Substituting \( \Delta \) into the formula for \( R \):
\[ R = \frac{a^3}{4 \cdot \frac{\sqrt{3}}{4} a^2} = \frac{a^3}{\sqrt{3} a^2} = \frac{a}{\sqrt{3}} \]

3. **Finding the Inradius \( r \)**:
- The inradius \( r \) is given by:
\[ r = \frac{\Delta}{s} \]
- The semi-perimeter \( s \) of the triangle is:
\[ s = \frac{a + a + a}{2} = \frac{3a}{2} \]
- Substituting \( \Delta \) and \( s \):
\[ r = \frac{\frac{\sqrt{3}}{4} a^2}{\frac{3a}{2}} = \frac{\sqrt{3} a^2}{4} \cdot \frac{2}{3a} = \frac{\sqrt{3} a}{6} \]

4. **Finding the Exradius \( r_1 \)**:
- The exradius \( r_1 \) opposite to side \( a \) is given by:
\[ r_1 = \frac{\Delta}{s - a} \]
- Here, \( s - a = \frac{3a}{2} - a = \frac{a}{2} \):
\[ r_1 = \frac{\frac{\sqrt{3}}{4} a^2}{\frac{a}{2}} = \frac{\sqrt{3} a^2}{4} \cdot \frac{2}{a} = \frac{\sqrt{3} a}{2} \]

5. **Finding the Ratio \( R : r : r_1 \)**:
- We have:
\[ R = \frac{a}{\sqrt{3}}, \quad r = \frac{\sqrt{3} a}{6}, \quad r_1 = \frac{\sqrt{3} a}{2} \]
- To find the ratio \( R : r : r_1 \):
\[ R : r : r_1 = \frac{a}{\sqrt{3}} : \frac{\sqrt{3} a}{6} : \frac{\sqrt{3} a}{2} \]
- Dividing each term by \( a \):
\[ = \frac{1}{\sqrt{3}} : \frac{\sqrt{3}}{6} : \frac{\sqrt{3}}{2} \]
- To eliminate the fractions, multiply through by \( 6\sqrt{3} \):
\[ = 6 : 3 : 9 \]
- Simplifying gives:
\[ = 2 : 1 : 3 \]

### Final Answer:
Thus, the ratio \( R : r : r_1 = 2 : 1 : 3 \).