We are given a quadrilateral $ABCD$. We need to find the intersection point $P$ between the side $AD$ and the fold line when folding the quadrilateral such that side $AB$ coincides with side $BC$. This fold line is the angle bisector of angle $ABC$. We need to construct point $P$ using a ruler and compass.

1. Problem Description

We are given a quadrilateral $ABCD$. We need to find the intersection point $P$ between the side $AD$ and the fold line when folding the quadrilateral such that side $AB$ coincides with side $BC$. This fold line is the angle bisector of angle $ABC$. We need to construct point $P$ using a ruler and compass.

2. Solution Steps

Step 1: Construct the angle bisector of angle $ABC$.

a. Place the compass point at vertex $B$ and draw an arc that intersects sides $AB$ and $BC$. Label the intersection points $X$ and $Y$ respectively.

b. Place the compass point at $X$ and draw an arc inside the angle $ABC$.

c. Place the compass point at $Y$ (using the same radius as in step b) and draw another arc inside the angle $ABC$. The two arcs should intersect. Label the intersection point $Z$.

d. Draw a line from $B$ to $Z$. The line $BZ$ is the angle bisector of angle $ABC$.

Step 2: Find the intersection point $P$ between the angle bisector $BZ$ and side $AD$.

a. Extend the angle bisector $BZ$ until it intersects with side $AD$.

b. The intersection point is the desired point $P$.

3. Final Answer

The point $P$ is the intersection of the angle bisector of angle $ABC$ and the side $AD$.