Suppose two convex quadrangles in the plane $P$ and $P'$, share a point $O$ such that, for every line $l$ trough $O$, the segment along which $l$ and $P$ meet is longer then the segment along which $l$ and $P'$ meet. Is it possible that the ratio of the area of $P'$ to the area of $P$ is greater then $1.9$?
1. **Define the quadrangles and the point \( O \)**:
   - Let \( P \) be a square \( ABCD \) with center \( O \).
   - Let \( P' \) be another convex quadrangle sharing the point \( O \).

2. **Construct \( P' \) with specific points**:
   - Pick points \( A' \in \overline{BA} \) and \( C' \in \overline{BC} \) such that \( AA' = CC' = \varepsilon_1 \), where \( \varepsilon_1 \) is a small positive number.
   - Construct \( P' \) as the image of \( OA'BC' \) under a homothety (scaling transformation) with a scale factor \( 2 - \varepsilon_2 \), where \( \varepsilon_2 \) is another small positive number.

3. **Calculate the area of \( P' \)**:
   - The area of \( P \) is \( [P] = \text{Area of } ABCD \).
   - The area of \( P' \) after the homothety is \( [P'] = (2 - \varepsilon_2)^2 \times \text{Area of } OA'BC' \).

4. **Compare the areas**:
   - For small enough \( \varepsilon_1 \) and \( \varepsilon_2 \), the area of \( P' \) can be made larger than \( 1.9 \times \text{Area of } P \).
   - Specifically, if \( \varepsilon_2 \) is small, \( (2 - \varepsilon_2)^2 \) can be made close to 4, and thus \( [P'] \) can be made significantly larger than \( [P] \).

5. **Verify the segment condition**:
   - For any line \( \ell \) through \( O \), it intersects \( P \) at two symmetric points \( T_1 \) and \( T_1' \) such that \( T_1 \) lies on the polyline \( A'BC' \).
   - The line \( \ell \) intersects \( P' \) at \( O \) and another point \( T_2 \).
   - The ratio \( \frac{T_1T_1'}{OT_2} = \frac{2OT_1}{OT_2} = \frac{2}{2 - \varepsilon_2} > 1 \), ensuring that the segment along which \( \ell \) and \( P \) meet is longer than the segment along which \( \ell \) and \( P' \) meet.

Thus, it is possible for the ratio of the area of \( P' \) to the area of \( P \) to be greater than 1.9.

The final answer is \( \boxed{\text{True}} \)