Cross ratio. Harmonic conjugates. Perspectivity. Projectivity
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 \[ \mathcal R(A_1,B_1;C_1,D_1)= \mathcal R(A_2,B_2;C_2,D_2).\quad\quad\quad\quad\quad (2)\] Thus it is meaningful to define the cross ratio of the pairs of concurrent points as \[ \mathcal R(a,b;c,d)=\mathcal R(A_1,B_1;C_1,D_1). \quad\quad\quad\quad\quad (3)\] 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 crossratio 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 \[ \mathcal R(A,B;C,D)=\mathcal R(A^*,B^*;C^*,D^*). \quad\quad\quad\quad\quad (6)\]
According to the previous statements perspectivity preserves the cross ratio and hence the harmonic conjugates.

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