If the length of tangents from `A, B, C` to the incircle of `Delta ABC` are `4,6,8` then which of the following is(are) correct? (All symbols used have usual meaning in a triangle.

Options

A. Area of `DeltaABC is 12sqrt6`
B. `r_(1), r_(2), r _(3)` are in HP
C. a,b,c are in AP
D. `r= (4 sqrt6)/(3)`

Answer

To solve the problem, we need to analyze the given lengths of the tangents from points A, B, and C to the incircle of triangle ABC. The lengths of the tangents are given as follows:

- Length of tangent from A = 4
- Length of tangent from B = 6
- Length of tangent from C = 8

Let’s denote the sides opposite to vertices A, B, and C as follows:
- Side opposite A (BC) = a
- Side opposite B (AC) = b
- Side opposite C (AB) = c

From the properties of tangents to a circle, we know that:
- \( s - a = 4 \)
- \( s - b = 6 \)
- \( s - c = 8 \)

Where \( s \) is the semi-perimeter of triangle ABC.

### Step 1: Find the semi-perimeter \( s \)

Using the equations for the tangents:
1. \( s - a = 4 \) → \( a = s - 4 \)
2. \( s - b = 6 \) → \( b = s - 6 \)
3. \( s - c = 8 \) → \( c = s - 8 \)

### Step 2: Express \( a, b, c \) in terms of \( s \)

Now we can express the sides \( a, b, c \) in terms of \( s \):
- \( a = s - 4 \)
- \( b = s - 6 \)
- \( c = s - 8 \)

### Step 3: Find the semi-perimeter \( s \)

The semi-perimeter \( s \) is given by:
\[ s = \frac{a + b + c}{2} \]

Substituting the expressions for \( a, b, c \):
\[ s = \frac{(s - 4) + (s - 6) + (s - 8)}{2} \]
\[ s = \frac{3s - 18}{2} \]
Multiplying both sides by 2:
\[ 2s = 3s - 18 \]
Rearranging gives:
\[ s = 18 \]

### Step 4: Calculate the sides \( a, b, c \)

Now substituting \( s = 18 \) back into the equations for \( a, b, c \):
- \( a = 18 - 4 = 14 \)
- \( b = 18 - 6 = 12 \)
- \( c = 18 - 8 = 10 \)

### Step 5: Check if \( a, b, c \) are in Arithmetic Progression (AP)

To check if \( a, b, c \) are in AP:
- The common difference \( d = b - a = 12 - 14 = -2 \)
- The common difference \( d = c - b = 10 - 12 = -2 \)

Since both differences are equal, \( a, b, c \) are in AP.

### Step 6: Calculate the area of triangle ABC

Using Heron's formula:
\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values:
\[ \text{Area} = \sqrt{18 \times (18 - 14) \times (18 - 12) \times (18 - 10)} \]
\[ = \sqrt{18 \times 4 \times 6 \times 8} \]
Calculating:
\[ = \sqrt{18 \times 192} = \sqrt{3456} = 12\sqrt{6} \]

### Step 7: Check if \( R_1, R_2, R_3 \) are in Harmonic Progression (HP)

Using the formulas:
- \( R_1 = \frac{A}{s - a} = \frac{12\sqrt{6}}{4} = 3\sqrt{6} \)
- \( R_2 = \frac{A}{s - b} = \frac{12\sqrt{6}}{6} = 2\sqrt{6} \)
- \( R_3 = \frac{A}{s - c} = \frac{12\sqrt{6}}{8} = \frac{3\sqrt{6}}{2} \)

To check if \( R_1, R_2, R_3 \) are in HP, we need to check if \( \frac{1}{R_1}, \frac{1}{R_2}, \frac{1}{R_3} \) are in AP:
- \( \frac{1}{R_1} = \frac{1}{3\sqrt{6}} \)
- \( \frac{1}{R_2} = \frac{1}{2\sqrt{6}} \)
- \( \frac{1}{R_3} = \frac{2}{3\sqrt{6}} \)

Checking if \( 2R_2 = R_1 + R_3 \):
- \( 2 \times \frac{1}{2\sqrt{6}} = \frac{1}{3\sqrt{6}} + \frac{2}{3\sqrt{6}} \)
- \( \frac{1}{\sqrt{6}} = \frac{3}{3\sqrt{6}} \)

Thus, \( R_1, R_2, R_3 \) are in HP.

### Conclusion

The correct options are:
1. Area of triangle ABC is \( 12\sqrt{6} \) (True)
2. \( R_1, R_2, R_3 \) are in HP (True)
3. \( a, b, c \) are in AP (True)
4. \( R = \frac{4\sqrt{6}}{3} \) (True)