Let A, B, C, A′, B′, C′, X, Y, Z, X′, Y′, Z′ and P be pairwise distinct points in space such that
A′∈BC; B′∈CA; C′∈AB; X′∈YZ; Y′∈ZX; Z′∈XY;
P∈AX; P∈BY; P∈CZ; P∈A′X′; P∈B′Y′; P∈C′Z′.
Prove that
\frac {BA^{\prime}}{A^{\prime}C}\cdot\frac {CB^{\prime}}{B^{\prime}A}\cdot\frac {AC^{\prime}}{C^{\prime}B} \equal{} \frac {YX^{\prime}}{X^{\prime}Z}\cdot\frac {ZY^{\prime}}{Y^{\prime}X}\cdot\frac {XZ^{\prime}}{Z^{\prime}Y}. invariantratiogeometry theoremsgeometry