In a `triangle ABC ,`the HM of the ex-radii is equal to

Options

A. 3r
B. 2R
C. R + r
D. none of these

Answer

To find the harmonic mean of the ex-radii \( r_1, r_2, r_3 \) of triangle \( ABC \), we can follow these steps:

### Step 1: Understand the Ex-Radii
The ex-radii \( r_1, r_2, r_3 \) correspond to the three vertices of triangle \( ABC \). They are defined as the radius of the excircle opposite to each vertex.

### Step 2: Formula for Harmonic Mean
The harmonic mean (HM) of three numbers \( r_1, r_2, r_3 \) is given by the formula:
\[ HM = \frac{3}{\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3}} \]

### Step 3: Use the Relationship Between Ex-Radii and In-Radius
There is a standard result that relates the ex-radii to the in-radius \( r \):
\[ \frac{1}{r} = \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} \]
This means that the sum of the reciprocals of the ex-radii can be expressed in terms of the in-radius.

### Step 4: Substitute the Relationship into the HM Formula
Using the relationship from Step 3, we can rewrite the harmonic mean:
\[ HM = \frac{3}{\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3}} = \frac{3}{\frac{1}{r}} = 3r \]

### Step 5: Conclusion
Thus, the harmonic mean of the ex-radii \( r_1, r_2, r_3 \) is equal to \( 3r \), where \( r \) is the in-radius of triangle \( ABC \).

### Final Answer
The harmonic mean of the ex-radii is \( 3r \).

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