MathDB
"Triple-ratios" are also projectively invariant

Source: 5th QEDMO problem 8, proposed by me but apparently known before

November 10, 2007
invariantratiogeometry theoremsgeometry

Problem Statement

Let A A, B B, C C, A′ A^{\prime}, B′ B^{\prime}, C′ C^{\prime}, X X, Y Y, Z Z, X′ X^{\prime}, Y′ Y^{\prime}, Z′ Z^{\prime} and P P be pairwise distinct points in space such that A′∈BC; B′∈CA; C′∈AB; X′∈YZ; Y′∈ZX; Z′∈XY; A^{\prime} \in BC;\ B^{\prime}\in CA;\ C^{\prime}\in AB;\ X^{\prime}\in YZ;\ Y^{\prime}\in ZX;\ Z^{\prime}\in XY; P∈AX; P∈BY; P∈CZ; P∈A′X′; P∈B′Y′; P∈C′Z′ P \in AX;\ P\in BY;\ P\in CZ;\ P\in A^{\prime}X^{\prime};\ P\in B^{\prime}Y^{\prime};\ P\in C^{\prime}Z^{\prime}. 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}.
Back to ProblemsView on AoPS