In an equilateral triangle, the inradius, circumradius, and one of the exradii are in the ratio

Options

A. `2: 4:5`
B. `1:2:3`
C. `1:2:4`
D. `2:4:3`

Explanation

To solve the problem of finding the ratio of the inradius, circumradius, and one of the exradii of an equilateral triangle, we can follow these steps:

### Step 1: Define the properties of the equilateral triangle
Let the side length of the equilateral triangle be \( a \). In an equilateral triangle, all sides are equal, and we can denote the sides as \( a = b = c \).

### Step 2: Calculate the area of the triangle
The area \( \Delta \) of an equilateral triangle is given by the formula:
\[ \Delta = \frac{\sqrt{3}}{4} a^2 \]

### Step 3: Calculate the semi-perimeter \( S \)
The semi-perimeter \( S \) of the triangle is given by:
\[ S = \frac{a + b + c}{2} = \frac{3a}{2} \]

### Step 4: Calculate the inradius \( r \)
The inradius \( r \) can be calculated using the formula:
\[ r = \frac{\Delta}{S} \]
Substituting the values from Steps 2 and 3:
\[ r = \frac{\frac{\sqrt{3}}{4} a^2}{\frac{3a}{2}} = \frac{\sqrt{3} a^2}{4} \cdot \frac{2}{3a} = \frac{a}{2\sqrt{3}} \]

### Step 5: Calculate the circumradius \( R \)
The circumradius \( R \) can be calculated using the formula:
\[ R = \frac{abc}{4\Delta} \]
For an equilateral triangle, this simplifies to:
\[ R = \frac{a \cdot a \cdot a}{4 \cdot \frac{\sqrt{3}}{4} a^2} = \frac{a^3}{\sqrt{3} a^2} = \frac{a}{\sqrt{3}} \]

### Step 6: Calculate one of the exradii \( r_1 \)
The exradius \( r_1 \) can be calculated using the formula:
\[ r_1 = \frac{\Delta}{S - a} \]
Substituting the values:
\[ r_1 = \frac{\frac{\sqrt{3}}{4} a^2}{\frac{3a}{2} - a} = \frac{\frac{\sqrt{3}}{4} a^2}{\frac{a}{2}} = \frac{\sqrt{3}}{2} a \]

### Step 7: Find the ratio of \( r : R : r_1 \)
Now we have:
- \( r = \frac{a}{2\sqrt{3}} \)
- \( R = \frac{a}{\sqrt{3}} \)
- \( r_1 = \frac{\sqrt{3}}{2} a \)

To find the ratio, we can express them in terms of \( a \):
\[ r : R : r_1 = \frac{a}{2\sqrt{3}} : \frac{a}{\sqrt{3}} : \frac{\sqrt{3}}{2} a \]
Dividing each term by \( a \):
\[ \frac{1}{2\sqrt{3}} : \frac{1}{\sqrt{3}} : \frac{\sqrt{3}}{2} \]

### Step 8: Simplify the ratio
To simplify:
1. Multiply each term by \( 2\sqrt{3} \):
\[ 1 : 2 : 3 \]

Thus, the ratio of the inradius, circumradius, and one of the exradii in an equilateral triangle is:
\[ 1 : 2 : 3 \]

### Final Answer
The correct option is \( 1 : 2 : 3 \).