Number of triangles `A B C` if `tanA=x ,tanB=x+1,a n dtanC=1-x` is ________

Explanation

To solve the problem of finding the number of triangles \( ABC \) given the conditions \( \tan A = x \), \( \tan B = x + 1 \), and \( \tan C = 1 - x \), we can use the property that in a triangle, the sum of the tangents of the angles is equal to the product of the tangents of the angles:

\[ \tan A + \tan B + \tan C = \tan A \cdot \tan B \cdot \tan C \]

### Step 1: Set up the equation

Substituting the given values into the equation:

\[ x + (x + 1) + (1 - x) = x \cdot (x + 1) \cdot (1 - x) \]

### Step 2: Simplify the left side

Combine the terms on the left side:

\[ x + x + 1 + 1 - x = 2 + x \]

So, the left side simplifies to:

\[ 2 + x \]

### Step 3: Simplify the right side

Now, simplify the right side:

\[ x \cdot (x + 1) \cdot (1 - x) = x \cdot (x + 1 - x^2 - x) = x \cdot (1 - x^2) \]

This simplifies to:

\[ x - x^3 \]

### Step 4: Set the equation

Now we have:

\[ 2 + x = x - x^3 \]

### Step 5: Rearrange the equation

Rearranging gives:

\[ x^3 - 2 = 0 \]

### Step 6: Solve for \( x \)

This can be rewritten as:

\[ x^3 = 2 \]

Taking the cube root of both sides:

\[ x = \sqrt[3]{2} \]

### Step 7: Find the angles

Now we can find the values of \( \tan A \), \( \tan B \), and \( \tan C \):

- \( \tan A = \sqrt[3]{2} \)
- \( \tan B = \sqrt[3]{2} + 1 \)
- \( \tan C = 1 - \sqrt[3]{2} \)

### Step 8: Determine the nature of the angles

Next, we need to check if any of these angles can be obtuse.

1. If \( \tan A = \sqrt[3]{2} \), then \( A \) is acute.
2. If \( \tan B = \sqrt[3]{2} + 1 \), this is also positive, so \( B \) is acute.
3. If \( \tan C = 1 - \sqrt[3]{2} \), since \( \sqrt[3]{2} \) is approximately \( 1.26 \), \( 1 - \sqrt[3]{2} \) is negative, which means \( C \) is obtuse.

### Step 9: Conclusion

Since we have one obtuse angle and two acute angles, a triangle can exist with these angle measures. However, we must check if two obtuse angles can exist, which they cannot.

Thus, the number of triangles \( ABC \) that can be formed under these conditions is:

\[ \text{Number of triangles} = 0 \]