$ABCD$ is a square. Describe the locus of points $P$, different from $A, B, C, D$, on that plane for which
\[\widehat{APB}+\widehat{CPD}=180^\circ\]
1. **Initial Observations**:
   - Let \(ABCD\) be a square with vertices \(A, B, C, D\) in the coordinate plane.
   - We need to find the locus of points \(P\) such that \(\angle APB + \angle CPD = 180^\circ\).

2. **Disk Observation**:
   - If \(\Gamma_1\) and \(\Gamma_2\) are the closed disks with diameters \(AB\) and \(CD\) respectively, then \(P \in \Gamma_1 \cup \Gamma_2\). Otherwise, the sum \(\angle APB + \angle CPD\) would be less than \(180^\circ\).

3. **Coordinate Setup**:
   - Assume \(A, B, C, D\) have coordinates \((0,1), (1,1), (1,0), (0,0)\) respectively.
   - Let \(P = (x, y)\).

4. **Case 1: \(P \in \Gamma_1\)**:
   - Reflect \(P\) over the line \(y = \frac{1}{2}\) to get a new point \(P'\).
   - By symmetry, \(\angle APB = \angle CP'D\).
   - Thus, quadrilateral \(CPDP'\) is cyclic.

5. **Cyclic Quadrilateral Condition**:
   - For \(CPDP'\) to be cyclic, the product of the slopes of the diagonals must be equal:
     \[
     y(y-1) = x(1-x)
     \]
   - This simplifies to:
     \[
     \left(x - \frac{1}{2}\right)^2 + \left(y - \frac{1}{2}\right)^2 = \left(\frac{1}{\sqrt{2}}\right)^2
     \]
   - This represents a circle centered at \(\left(\frac{1}{2}, \frac{1}{2}\right)\) with radius \(\frac{1}{\sqrt{2}}\).

6. **Case 2: \(P \in \Gamma_2\)**:
   - Define \(P' = (x, y-1)\).
   - Again, by symmetry, \(\angle APB = \angle CP'D\).
   - Thus, quadrilateral \(CPDP'\) is cyclic.

7. **Cyclic Quadrilateral Condition**:
   - For \(CPDP'\) to be cyclic, the product of the slopes of the diagonals must be equal:
     \[
     y(1-y) = x(1-x)
     \]
   - This simplifies to:
     \[
     x = y \quad \text{or} \quad x + y = 1
     \]

8. **Conclusion**:
   - The locus of points \(P\) is the union of the arcs \(AB\) and \(CD\) of the circumcircle and the two diagonals of the square, except for the vertices \(A, B, C, D\).

The final answer is the union of the arcs \( \boxed{ AB } \) and \(CD\) of the circumcircle and the two diagonals of the square, except for the vertices \(A, B, C, D\).