To solve the problem, we need to analyze the conditions given for the complex numbers \( a, b, c \).
### Step-by-step Solution:
1. **Understanding the Conditions**:
We have three complex numbers \( a, b, c \) such that:
\[
|a| = |b| = |c| = 1
\]
This means that \( a, b, c \) lie on the unit circle in the complex plane.
2. **Using the Given Equations**:
We also know:
\[
a + b + c = 1
\]
\[
abc = 1
\]
3. **Expressing \( a, b, c \)**:
Since \( |a| = |b| = |c| = 1 \), we can express these complex numbers in exponential form:
\[
a = e^{i\theta_1}, \quad b = e^{i\theta_2}, \quad c = e^{i\theta_3}
\]
where \( \theta_1, \theta_2, \theta_3 \) are angles corresponding to points on the unit circle.
4. **Substituting into the Equations**:
The equation \( abc = 1 \) implies:
\[
e^{i(\theta_1 + \theta_2 + \theta_3)} = 1
\]
This means:
\[
\theta_1 + \theta_2 + \theta_3 = 2k\pi \quad \text{for some integer } k
\]
5. **Using the Sum Equation**:
From \( a + b + c = 1 \), we can rewrite it as:
\[
e^{i\theta_1} + e^{i\theta_2} + e^{i\theta_3} = 1
\]
6. **Finding the Roots**:
We can consider the polynomial whose roots are \( a, b, c \):
\[
P(z) = z^3 - (a+b+c)z^2 + (ab + ac + bc)z - abc
\]
Substituting the known values:
\[
P(z) = z^3 - z^2 + (ab + ac + bc)z - 1
\]
7. **Finding \( ab + ac + bc \)**:
From the identity \( |a|^2 + |b|^2 + |c|^2 = 3 \) and \( a + b + c = 1 \), we can derive:
\[
ab + ac + bc = \frac{(a+b+c)^2 - (|a|^2 + |b|^2 + |c|^2)}{2} = \frac{1^2 - 3}{2} = -1
\]
8. **Final Polynomial**:
Thus, we have:
\[
P(z) = z^3 - z^2 - z - 1 = 0
\]
9. **Finding the Roots of the Polynomial**:
We can factor this polynomial or use numerical methods to find the roots. The roots are:
\[
z = 1, \quad z = i, \quad z = -i
\]
Therefore, we can assign:
\[
a = i, \quad b = -i, \quad c = 1
\]
10. **Conclusion**:
The points \( A, B, C \) in the complex plane correspond to the coordinates \( (0, 1), (0, -1), (1, 0) \) respectively. Since two of the points are symmetric about the real axis, triangle \( ABC \) is isosceles.
