The sides of a triangle are three consecutive numbers and its largest angle is twice the smallest one. Find the sides of the triangle.

Answer

To solve the problem step by step, we will denote the sides of the triangle as three consecutive integers. Let's denote the sides as \( n-1 \), \( n \), and \( n+1 \), where \( n \) is a positive integer. The largest angle is opposite the longest side, which is \( n+1 \), and the smallest angle is opposite the shortest side, which is \( n-1 \).

### Step 1: Set up the angles
Let:
- Angle \( A \) be opposite side \( n-1 \)
- Angle \( B \) be opposite side \( n \)
- Angle \( C \) be opposite side \( n+1 \)

According to the problem, angle \( C \) (the largest angle) is twice angle \( A \):
\[ C = 2A \]

### Step 2: Use the Sine Rule
By the sine rule:
\[ \frac{\sin A}{n-1} = \frac{\sin C}{n+1} \]
Substituting \( C = 2A \) into the equation gives:
\[ \frac{\sin A}{n-1} = \frac{\sin 2A}{n+1} \]

### Step 3: Use the double angle formula for sine
Using the double angle formula, \( \sin 2A = 2 \sin A \cos A \):
\[ \frac{\sin A}{n-1} = \frac{2 \sin A \cos A}{n+1} \]

### Step 4: Simplify the equation
Assuming \( \sin A \neq 0 \), we can divide both sides by \( \sin A \):
\[ \frac{1}{n-1} = \frac{2 \cos A}{n+1} \]

### Step 5: Cross-multiply
Cross-multiplying gives:
\[ (n+1) = 2(n-1) \cos A \]
This simplifies to:
\[ n + 1 = 2n \cos A - 2 \cos A \]
Rearranging gives:
\[ 2n \cos A - n = 2 \cos A - 1 \]

### Step 6: Isolate \( \cos A \)
Rearranging terms gives:
\[ \cos A (2n - 2) = n + 1 \]
Thus:
\[ \cos A = \frac{n + 1}{2(n - 1)} \]

### Step 7: Apply the Cosine Rule
Using the cosine rule for angle \( A \):
\[ \cos A = \frac{(n^2 + (n+1)^2 - (n-1)^2)}{2n(n+1)} \]

### Step 8: Substitute and simplify
Substituting the expressions for \( \cos A \):
\[ \frac{(n^2 + (n^2 + 2n + 1) - (n^2 - 2n + 1))}{2n(n+1)} = \frac{(n^2 + n^2 + 2n + 1 - n^2 + 2n - 1)}{2n(n+1)} \]
This simplifies to:
\[ \frac{(n^2 + 4n)}{2n(n+1)} = \frac{n + 1}{2(n - 1)} \]

### Step 9: Cross-multiply again
Cross-multiplying gives:
\[ (n^2 + 4n)(n - 1) = (n + 1)(2n(n + 1)) \]

### Step 10: Expand and simplify
Expanding both sides leads to a quadratic equation. Solving this will yield the value of \( n \).

### Step 11: Solve the quadratic equation
After simplifying, we find \( n = 5 \).

### Step 12: Find the sides of the triangle
The sides of the triangle are:
- \( n-1 = 4 \)
- \( n = 5 \)
- \( n+1 = 6 \)

Thus, the sides of the triangle are 4, 5, and 6.