1964 IMO 第 6 题

Given a tetrahedron $A B C D$, let $D_{1}$ be the centroid of the triangle $A B C$ and let $A_{1}, B_{1}, C_{1}$ be the intersection points of the lines parallel to $D D_{1}$ and passing through the points $A, B, C$ with the opposite faces of the tetrahedron. Prove that the volume of the tetrahedron $A B C D$ is onethird the volume of the tetrahedron $A_{1} B_{1} C_{1} D_{1}$. Does the result remain true if the point $D_{1}$ is replaced with any point inside the triangle $A B C$ ?

给定一个四面体 $A B C D$，设 $D_{1}$ 为三角形 $A B C$ 的质心，设 $A_{1}、B_{1}、C_{1}$ 为平行于 $D D_{1}$ 并穿过点 $A、B、C$ 与四面体相对面的直线的交点。证明四面体$A B C D$的体积是四面体$A_{1} B_{1} C_{1} D_{1}$体积的三分之一。如果将点 $D_{1}$ 替换为三角形 $A B C$ 内的任意点，结果是否仍然为真？