Find the locus of the vertex of a triangle whose, sum of base angles is fixed.

Answer

To find the locus of the vertex of a triangle whose sum of base angles is fixed, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Triangle**:
- Let the triangle be \( ABC \) with \( A \) and \( B \) as the base points on the x-axis, and \( C \) as the vertex above the x-axis.
- Let the coordinates of \( A \) be \( (-a, 0) \) and \( B \) be \( (a, 0) \). The vertex \( C \) will have coordinates \( (h, k) \).

2. **Base Angles**:
- Let the angles at \( A \) and \( B \) be \( \phi_1 \) and \( \phi_2 \) respectively. According to the problem, the sum of these angles is fixed:
\[ \phi_1 + \phi_2 = \text{constant} \]

3. **Using Tangent Function**:
- From triangle \( ABC \), we can express the tangents of the angles:
\[ \tan(\phi_1) = \frac{k}{h + a} \quad \text{(from point A)} \]
\[ \tan(\phi_2) = \frac{k}{a - h} \quad \text{(from point B)} \]

4. **Relating the Angles**:
- Since \( \phi_2 = \text{constant} - \phi_1 \), we can write:
\[ \tan(\phi_2) = \tan(\text{constant} - \phi_1) \]
- Using the tangent subtraction formula:
\[ \tan(\phi_2) = \frac{\tan(\text{constant}) - \tan(\phi_1)}{1 + \tan(\text{constant})\tan(\phi_1)} \]

5. **Setting Up the Equation**:
- Equating the two expressions for \( \tan(\phi_2) \):
\[ \frac{k}{a - h} = \frac{\tan(\text{constant}) - \tan(\phi_1)}{1 + \tan(\text{constant})\tan(\phi_1)} \]

6. **Substituting \( \tan(\phi_1) \)**:
- Substitute \( \tan(\phi_1) = \frac{k}{h + a} \) into the equation:
\[ \frac{k}{a - h} = \frac{\tan(\text{constant}) - \frac{k}{h + a}}{1 + \tan(\text{constant})\frac{k}{h + a}} \]

7. **Cross Multiplying and Rearranging**:
- Cross-multiply to eliminate the fractions and rearrange the terms to isolate \( k \) in terms of \( h \):
\[ k(1 + \tan(\text{constant})\frac{k}{h + a}) = (a - h)(\tan(\text{constant}) - \frac{k}{h + a}) \]

8. **Finding the Locus**:
- After simplifying, we will arrive at a quadratic equation in terms of \( h \) and \( k \). This will represent the locus of point \( C \).

9. **Final Form**:
- The final equation will typically be of the form:
\[ k^2 = 4p(h - h_0) \]
- This represents a parabola.

### Conclusion:
The locus of the vertex \( C \) of the triangle, whose sum of base angles is fixed, is a parabola.