# 3. CONDITION

Given a right triangle $A B C$ with legs $A C=3$ and $B C=4$. Construct triangle $A_{1} B_{1} C_{1}$ by sequentially moving point $A$ a certain distance parallel to segment $B C$ (point $A_{1}$ ), then point $B-$ parallel to segment $A_{1} C$ (point $B_{1}$ ), and finally point $C$ - parallel to segment $A_{1} B_{1}$ (point $C_{1}$ ). What is the length of segment $B_{1} C_{1}$, if it turns out that angle $A_{1} B_{1} C_{1}$ is a right angle and $A_{1} B_{1}=1$?
Solution. When a vertex of a triangle is moved parallel to its base, the area of the triangle does not change. Therefore, we sequentially obtain the equality of the areas of triangles $A B C, A_{1} B C, A_{1} B_{1} C$, and finally, $A_{1} B_{1} C_{1}$. Thus, $B_{1} C_{1}=2 S / A_{1} B_{1}=12$.

Answer: 12.