Cross ratio. Harmonic conjugates. Perspectivity. Projectivity

Definition
 
Let \( A \), \( B \), \( C \), and \( D \) be colinear points. The cross ratio of the pairs of points \( (A,B) \) and \( (C,D) \) is \[ \mathcal{R}(A,B;C,D) = \frac{\overrightarrow{AC}} {\overrightarrow{CB}}:\frac{\overrightarrow{AD}} {\overrightarrow{DB}}. \quad\quad\quad\quad\quad (1) \]

Let \( a \), \( b \), \( c \), \( d \) be four concurrent lines. For the given lines \( p_1 \) and \( p_2 \) let us denote \( A_i=a\cap p_i \), \( B_i=b\cap p_i \), \( C_i=c\cap p_i \), \( D_i=d\cap p_i \), for \( i=1 \), \( 2 \). Then \[ \mathcal R(A_1,B_1;C_1,D_1)= \mathcal R(A_2,B_2;C_2,D_2).\quad\quad\quad\quad\quad (2)\]

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Thus it is meaningful to define the cross ratio of the pairs of concurrent points as \[ \mathcal R(a,b;c,d)=\mathcal R(A_1,B_1;C_1,D_1). \quad\quad\quad\quad\quad (3)\]

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Assume that points \( O_1 \), \( O_2 \), \( A \), \( B \), \( C \), \( D \) belong to a circle. Then \[ \mathcal R(O_1A,O_1B;O_1C,O_1D) = \mathcal R(O_2A,O_2B;O_2C,O_2D).\quad\quad\quad\quad\quad (4)\] Hence it is meaningful to define the cross-ratio for cocyclic points as \[ \mathcal R(A,B;C,D) =\mathcal R(O_1A,O_1B;O_1C,O_1D).\quad\quad\quad\quad\quad (5)\] Assume that the points \( A \), \( B \), \( C \), \( D \) are colinear or cocyclic. Let an inversion with center \( O \) maps \( A \), \( B \), \( C \), \( D \) into \( A^* \), \( B^* \), \( C^* \), \( D^* \). Then \[ \mathcal R(A,B;C,D)=\mathcal R(A^*,B^*;C^*,D^*). \quad\quad\quad\quad\quad (6)\]

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Definition
 
Assume that \( A \), \( B \), \( C \), and \( D \) are cocyclic or colinear points. Pairs of points \( (A,B) \) and \( (C,D) \) are harmonic conjugates if \( \mathcal R(A,B;C,D)=-1 \). We also write \( \mathcal H(A,B;C,D) \) when we want to say that \( (A,B) \) and \( (C,D) \) are harmonic conjugates to each other.

Definition
 
Let each of \( l_1 \) and \( l_2 \) be either line or circle. Perspectivity with respect to the point \( S \) (denote as \( \frac{S}{\overline\wedge} \)), is the mapping of \( l_1\rightarrow l_2 \), such that

(i) If either \( l_1 \) or \( l_2 \) is a circle than it contains \( S \);

(ii) every point \( A_1\in l_1 \) is mapped to the point \( A_2=OA_1\cap l_2 \).

According to the previous statements perspectivity preserves the cross ratio and hence the harmonic conjugates.

Definition
 
Let each of \( l_1 \) and \( l_2 \) be either line or circle. Projectivity is any mapping from \( l_1 \) to \( l_2 \) that can be represented as a finite composition of perspectivities.

Theorem 1
 
Assume that the points \( A \), \( B \), \( C \), \( D_1 \), and \( D_2 \) are either colinear or cocyclic. If the equation \( \mathcal R(A,B;C,D_1)= \mathcal R(A,B;C,D_2) \) is satisfied, then \( D_1=D_2 \). In other words, a projectivity with three fixed points is the identity.

Theorem 2
 
If the points \( A \), \( B \), \( C \), \( D \) are mutually discjoint and \( \mathcal R(A,B;C,D)=\mathcal R(B,A;C,D) \) then \( \mathcal H(A,B;C,D) \).


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