Iranian Mathematical Olympiad

Let quadrilateral that these four pieces form in the space be $ABCD$ and we marked points $M,N,P$ and $Q$ on sides $AB,BC,CD$ and $DA$ respectively, so that the marked points are on a plane named $\pi$. Let $a,b,c$ and $d$ be the distances from $A,B,C$ and $D$ to $\pi$ respectively. So we have



$$\frac{AM}{MB}=\frac{a}{b},\frac{BN}{NC}=\frac{b}{c},\frac{CP}{PD}=\frac{c}{d},\frac{DQ}{QA}=\frac{d}{a}\Rightarrow \frac{AM}{MB}\times \frac{BN}{NC}\times \frac{CP}{PD}\times \frac{DQ}{QA}=1.$$

Now $ABCD$ varies, let ${\pi }^{\prime }$ be the plane passing through $M,N,P$. Suppose that $a^{\prime },b^{\prime },c^{\prime }$ and $d^{\prime }$ are the distances of $A,B,C$ and $D$ to ${\pi }^{\prime }$ respectively. Thus

$$\frac{AM}{MB}=\frac{a^{\prime }}{b^{\prime }},\frac{BN}{NC}=\frac{b^{\prime }}{c^{\prime }},\frac{CP}{PD}=\frac{c^{\prime }}{d^{\prime }}\Rightarrow \frac{DQ}{QA}=\frac{d^{\prime }}{a^{\prime }}.$$

So $Q$ must be on ${\pi }^{\prime }$ too.