To solve the problem of determining the position of the center of mass (COM) in a system of four unequal particles arranged in a non-linear coplanar configuration, we can follow these steps:
### Step-by-Step Solution:
1. **Understanding the Concept of Center of Mass**:
The center of mass of a system of particles is the point where the total mass of the system can be considered to be concentrated. For a system of particles, the center of mass can be calculated using the formula:
\[
\text{COM} = \frac{\sum m_i \cdot r_i}{\sum m_i}
\]
where \( m_i \) is the mass of each particle and \( r_i \) is the position vector of each particle.
2. **Identifying the Particles**:
Let's denote the four particles as \( P_1, P_2, P_3, \) and \( P_4 \) with masses \( m_1, m_2, m_3, \) and \( m_4 \) respectively. The positions of these particles in a 2D plane can be represented as coordinates \( (x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4) \).
3. **Calculating the Center of Mass**:
Using the formula for the center of mass, we can find the coordinates of the COM:
\[
x_{COM} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3 + m_4 x_4}{m_1 + m_2 + m_3 + m_4}
\]
\[
y_{COM} = \frac{m_1 y_1 + m_2 y_2 + m_3 y_3 + m_4 y_4}{m_1 + m_2 + m_3 + m_4}
\]
4. **Analyzing the Position of the Center of Mass**:
In a non-linear coplanar arrangement, the center of mass will not necessarily lie within the triangle formed by any three particles. However, it will lie within the convex hull formed by all four particles. This means that the COM can be inside or on the edge of the triangle formed by the heaviest three particles.
5. **Conclusion**:
Based on the arrangement of the particles and their masses, the center of mass will lie at a point that is influenced by the distribution of mass among the particles. It may lie within the triangle formed by the three heaviest particles or at the edge of that triangle.
### Final Answer:
The center of mass must lie within or at the edge of at least one of the triangles formed by any three of the four particles.
