Let $ABCD$ be a tetrahedron and $A'$, $B'$, $C'$ be arbitrary points on the edges $[DA]$, $[DB]$, $[DC]$, respectively. One considers the points $P_c \in [AB]$, $P_a \in [BC]$, $P_b \in [AC]$ and $P'_c \in [A'B']$, $P'_a \in [B'C']$, $P'_b \in [A'C']$ such that $$\frac{P_cA}{P_cB}= \frac{P'_cA'}{P'_cB'}=\frac{AA'}{BB'}\,\,\, , \,\,\,\frac{P_aB}{P_aC}= \frac{P'_aB'}{P'_aC'}=\frac{BB'}{CC'}\,\,\, , \,\,\, \frac{P_bC}{P_bA}= \frac{P'_bC'}{P'_bA'}=\frac{CC'}{AA'}$$Prove that: a) the lines $AP_a,$ $BP_b$, $CP_c$ have a common point $P$ and the lines $A'P'_a$, $B'P'_b$ , $C'P'_c$ have a common point $P'$ b) $\frac{PC}{PP_c}=\frac{P'C'}{P'P'_c} $ c) if $A', B', C'$ are variable points on the edges $[DA]$, $[DB]$, $[DC]$, then the line $PP'$ is always parallel to a fixed line.