$P$ is an interior point of the angle whose sides are the rays $OA$ and $OB.$ Locate $X$ on $OA$ and $Y$ on $OB$ so that the line segment $\overline{XY}$ contains $P$ and so that the product $(PX)(PY)$ is a minimum.
1. **Choose Points \(X\) and \(Y\)**:
   We need to locate points \(X\) on \(OA\) and \(Y\) on \(OB\) such that the line segment \(\overline{XY}\) contains \(P\) and the product \((PX)(PY)\) is minimized. We choose \(X\) and \(Y\) such that \(OX = OY\). This implies that \(XY\) is perpendicular to the bisector of \(\angle AOB\).

2. **Introduce Alternative Points \(X'\) and \(Y'\)**:
   Let \(X'\) and \(Y'\) be any other points on \(OA\) and \(OB\) respectively, such that \(P\) is on the segment \(\overline{X'Y'}\).

3. **Define Angles**:
   Let \(\alpha = \angle OYX = \angle YXO\), \(\beta = \angle OY'X'\), and \(\gamma = \angle Y'X'O\). Note that \(\beta + \gamma = 2\alpha = \pi - \angle XOY\).

4. **Apply the Law of Sines**:
   By the Law of Sines in triangles \(PXX'\) and \(PYY'\), we have:
   \[
   \frac{PX}{PX'} = \frac{\sin \gamma}{\sin \alpha} \quad \text{and} \quad \frac{PY}{PY'} = \frac{\sin \beta}{\sin \alpha}
   \]

5. **Product of Segments**:
   We need to prove that \(\sin \beta \cdot \sin \gamma < \sin^2 \alpha\).

6. **Logarithmic Transformation**:
   Taking the logarithm, this reduces to:
   \[
   \frac{\log \sin \beta + \log \sin \gamma}{2} < \log \sin \frac{\beta + \gamma}{2}
   \]

7. **Concavity of \(\log \sin x\)**:
   The function \(\log \sin x\) is strictly concave on \((0, \pi)\), since its derivative \(\frac{\cos x}{\sin x} = \cot x\) is strictly decreasing. By Jensen's inequality for concave functions, we have:
   \[
   \frac{\log \sin \beta + \log \sin \gamma}{2} < \log \sin \left(\frac{\beta + \gamma}{2}\right)
   \]

8. **Conclusion**:
   Since \(\beta + \gamma = 2\alpha\), we have:
   \[
   \log \sin \left(\frac{\beta + \gamma}{2}\right) = \log \sin \alpha
   \]
   Therefore:
   \[
   \frac{\log \sin \beta + \log \sin \gamma}{2} < \log \sin \alpha
   \]
   which implies:
   \[
   \sin \beta \cdot \sin \gamma < \sin^2 \alpha
   \]

Thus, choosing \(X\) and \(Y\) such that \(OX = OY\) indeed minimizes the product \((PX)(PY)\).

The final answer is \(\boxed{OX = OY}\).