"Triple-ratios" are also projectively invariant

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"Triple-ratios" are also projectively invariant

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

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#1 h Nov 10, 2007, 1:51 PM • 3 Y

Y by Adventure10, Mango247, and 1 other user

Let $A$ , $B$ , $C$ , $A^{\prime}$ , $B^{\prime}$ , $C^{\prime}$ , $X$ , $Y$ , $Z$ , $X^{\prime}$ , $Y^{\prime}$ , $Z^{\prime}$ and $P$ be pairwise distinct points in space such that

$A^{\prime} \in BC;\ B^{\prime}\in CA;\ C^{\prime}\in AB;\ X^{\prime}\in YZ;\ Y^{\prime}\in ZX;\ Z^{\prime}\in XY;$
$P \in AX;\ P\in BY;\ P\in CZ;\ P\in A^{\prime}X^{\prime};\ P\in B^{\prime}Y^{\prime};\ P\in C^{\prime}Z^{\prime}$ .

Prove that

$\frac {BA^{\prime}}{A^{\prime}C}\cdot\frac {CB^{\prime}}{B^{\prime}A}\cdot\frac {AC^{\prime}}{C^{\prime}B} = \frac {YX^{\prime}}{X^{\prime}Z}\cdot\frac {ZY^{\prime}}{Y^{\prime}X}\cdot\frac {XZ^{\prime}}{Z^{\prime}Y}$ .

This post has been edited 1 time. Last edited by darij grinberg, Apr 6, 2009, 12:26 PM

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#2 h Nov 10, 2007, 1:51 PM • 2 Y

Y by Adventure10 and 1 other user

Generalization and extension: Let $A_1$ , $A_2$ , ..., $A_n$ , $A_1^{\prime}$ , $A_2^{\prime}$ , ..., $A_n^{\prime}$ , $X_1$ , $X_2$ , ..., $X_n$ , $X_1^{\prime}$ , $X_2^{\prime}$ , ..., $X_n^{\prime}$ and $P$ be points in space such that for every $i\in\left\{1,2,...,n\right\}$ , we have

$A_i^{\prime}\in A_iA_{i + 1}$ ; $X_i^{\prime}\in X_iX_{i + 1}$ ; $P\in A_iX_i$ ; $P\in A_i^{\prime}X_i^{\prime}$ ; $A_i^{\prime}\neq A_i$ ; $A_i^{\prime}\neq A_{i + 1}$ ; $P\not\in A_iA_{i + 1}$ ; $P\not\in X_iX_{i + 1}$ ,

where the indices are cyclic modulo $n$ (so that $A_{n + 1}$ means the same as $A_1$ ).

Prove that

$\prod_{i = 1}^{n}\frac {A_iA_i^{\prime}}{A_i^{\prime}A_{i + 1}} = \prod_{i = 1}^{n}\frac {X_iX_i^{\prime}}{X_i^{\prime}X_{i + 1}}$ ,

where the segments are directed.

This generalization yields the above problem for $n = 3$ , and the invariance of the cross-ratio under central projection for $n = 2$ .

Darij

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