If P, Q, R, S are four coplanar points on the sides AB, BC, CD, DA of a skew quadrilateral, then `(AB)/(PB)cdot(BQ)/(QC)cdot(CR)/(RD)cdot(DS)/(SA)` equals

Options

A. `1`
B. `-1`
C. `3`
D. `-3`

Answer

To solve the problem, we need to find the value of the expression:

\[ \frac{AB}{PB} \cdot \frac{BQ}{QC} \cdot \frac{CR}{RD} \cdot \frac{DS}{SA} \]

where \(P\), \(Q\), \(R\), and \(S\) are coplanar points on the sides \(AB\), \(BC\), \(CD\), and \(DA\) of a skew quadrilateral.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let \(A\), \(B\), \(C\), and \(D\) be the vertices of the skew quadrilateral.
- The points \(P\), \(Q\), \(R\), and \(S\) lie on the segments \(AB\), \(BC\), \(CD\), and \(DA\) respectively.

2. **Defining Ratios**:
- Let:
- \( \lambda = \frac{AP}{PB} \)
- \( \mu = \frac{BQ}{QC} \)
- \( \nu = \frac{CR}{RD} \)
- \( \sigma = \frac{DS}{SA} \)

3. **Expressing Coordinates**:
- The coordinates of point \(P\) can be expressed as:
\[ P = \left( \frac{\lambda x_2 + x_1}{\lambda + 1}, \frac{\lambda y_2 + y_1}{\lambda + 1}, \frac{\lambda z_2 + z_1}{\lambda + 1} \right) \]
- Similarly, we can express the coordinates for points \(Q\), \(R\), and \(S\) using their respective ratios.

4. **Using the Plane Equation**:
- Since \(P\), \(Q\), \(R\), and \(S\) are coplanar, they satisfy the plane equation:
\[ Ax + By + Cz + D = 0 \]
- Substituting the coordinates of \(P\) into this equation gives us a relationship involving \(\lambda\).

5. **Setting Up Relationships**:
- By substituting the coordinates of \(P\), \(Q\), \(R\), and \(S\) into the plane equation, we can derive relationships between \(\lambda\), \(\mu\), \(\nu\), and \(\sigma\):
\[ \lambda = -\frac{L_1}{L_2}, \quad \mu = -\frac{L_2}{L_3}, \quad \nu = -\frac{L_3}{L_4}, \quad \sigma = -\frac{L_4}{L_1} \]
- Here, \(L_1\), \(L_2\), \(L_3\), and \(L_4\) are the values obtained from substituting the coordinates of points \(A\), \(B\), \(C\), and \(D\) into the plane equation.

6. **Combining the Ratios**:
- Now substituting these relationships into the original expression:
\[ \frac{AB}{PB} \cdot \frac{BQ}{QC} \cdot \frac{CR}{RD} \cdot \frac{DS}{SA} = \lambda \cdot \mu \cdot \nu \cdot \sigma \]
- This simplifies to:
\[ \lambda \cdot \mu \cdot \nu \cdot \sigma = \left(-\frac{L_1}{L_2}\right) \cdot \left(-\frac{L_2}{L_3}\right) \cdot \left(-\frac{L_3}{L_4}\right) \cdot \left(-\frac{L_4}{L_1}\right) = 1 \]

7. **Conclusion**:
- Thus, we conclude that:
\[ \frac{AB}{PB} \cdot \frac{BQ}{QC} \cdot \frac{CR}{RD} \cdot \frac{DS}{SA} = 1 \]

### Final Answer:
\[ \frac{AB}{PB} \cdot \frac{BQ}{QC} \cdot \frac{CR}{RD} \cdot \frac{DS}{SA} = 1 \]