Consider a circle with center $O$ and radius $R,$ and let $A$ and $B$ be two points in the plane of this circle.

[b]a.)[/b] Draw a chord $CD$ of the circle such that $CD$ is parallel to $AB,$ and the point of the intersection $P$ of the lines $AC$ and $BD$ lies on the circle.

[b]b.)[/b] Show that generally, one gets two possible points $P$ ($P_{1}$ and $P_{2}$) satisfying the condition of the above problem, and compute the distance between these two points, if the lengths $OA=a,$ $OB=b$ and $AB=d$ are given.
### Part (a)

1. **Draw the chord \(CD\) parallel to \(AB\):**
   - Given a circle \(\mathcal{C}(O, R)\) with center \(O\) and radius \(R\), and two points \(A\) and \(B\) in the plane of the circle.
   - Draw a chord \(CD\) such that \(CD \parallel AB\).
   - Let \(P\) be the point of intersection of the lines \(AC\) and \(BD\) such that \(P\) lies on the circle.

### Part (b)

1. **Homothety and Tangent Circles:**
   - Consider the homothety with center \(P\) that maps \(C \to A\) and \(D \to B\). This homothety transforms the circle \(\mathcal{C}(O, R)\) into another circle tangent to \(\mathcal{C}\) at \(P\) and passing through \(A\) and \(B\).
   - Conversely, if there is a circle tangent to \(\mathcal{C}\) and passing through \(A\) and \(B\), then the point of tangency \(P\) satisfies the conditions.

2. **Constructing Tangent Circles:**
   - To find the two possible points \(P\), we need to find the two circles tangent (internally and externally) to \(\mathcal{C}\) and passing through \(A\) and \(B\).
   - Perform an inversion with pole \(A\). Let \(B'\) be the image of \(B\) under this inversion.
   - Construct the image \(\mathcal{C}'\) of \(\mathcal{C}\) under this inversion.
   - Construct the two tangents from \(B'\) to \(\mathcal{C}'\). These tangents will intersect \(\mathcal{C}'\) at two points, which correspond to the two circles we are looking for when we perform the inversion again.

3. **Distance Between Points \(P_1\) and \(P_2\):**
   - Let \(P_1\) and \(P_2\) be the two points of tangency.
   - The distance between \(P_1\) and \(P_2\) can be computed using the given lengths \(OA = a\), \(OB = b\), and \(AB = d\).

4. **Using Power of a Point Theorem:**
   - The power of point \(A\) with respect to the circle is given by \(AP_1 \cdot AP_2 = R^2 - a^2\).
   - Similarly, the power of point \(B\) with respect to the circle is given by \(BP_1 \cdot BP_2 = R^2 - b^2\).

5. **Using the Chord Length Formula:**
   - The length of the chord \(P_1P_2\) can be found using the formula for the length of a chord in a circle:
     \[
     P_1P_2 = 2R \sin\left(\frac{\theta}{2}\right)
     \]
     where \(\theta\) is the angle subtended by the chord at the center of the circle.

6. **Calculating the Angle \(\theta\):**
   - Using the given lengths \(OA = a\), \(OB = b\), and \(AB = d\), we can find the angle \(\theta\) using the cosine rule in triangle \(OAB\):
     \[
     \cos(\theta) = \frac{a^2 + b^2 - d^2}{2ab}
     \]
   - Therefore, the angle \(\theta\) is:
     \[
     \theta = \cos^{-1}\left(\frac{a^2 + b^2 - d^2}{2ab}\right)
     \]

7. **Final Distance Calculation:**
   - Substitute \(\theta\) into the chord length formula to find \(P_1P_2\):
     \[
     P_1P_2 = 2R \sin\left(\frac{\cos^{-1}\left(\frac{a^2 + b^2 - d^2}{2ab}\right)}{2}\right)
     \]

The final answer is \(\boxed{2R \sin\left(\frac{\cos^{-1}\left(\frac{a^2 + b^2 - d^2}{2ab}\right)}{2}\right)}\)