# 3. CONDITION

Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge 1. A line $l$ passes through point $E$, the midpoint of edge $C_{1} D_{1}$, and intersects lines $A D_{1}$ and $A_{1} B$. Find the distance from point $E$ to the point of intersection of line $l$ with line $A_{1} B$.
Solution. Consider the plane $A D_{1} C_{1} B$. Since the line $l$ passes through the midpoint of the edge $C_{1} D_{1}$ and intersects the line $A D_{1}$, it has two common points with this plane, which means it entirely belongs to this plane. At the same time, the line $A_{1} B$ lies in the plane $A B A_{1} B_{1}$. Therefore, the intersection point of the lines $l$ and $A_{1} B$ lies on the line of intersection of the planes $A D_{1} C_{1} B$ and $A B A_{1} B_{1}$, i.e., on the line $A B$. But at the same time, it belongs to the line $A_{1} B$, which means it coincides with the intersection point of the lines $l$ and $A_{1} B$, i.e., with the point $B$. Thus, we need to find the distance between the points $E$ and $B$. By the Pythagorean theorem: $B E=$ $\sqrt{1+1+1 / 4}=1.5$

Answer. 1.5.