The Harmonic Construction

Given 4 points (A,B,C,D) on a line in that order, their cross ratio is defined as


Textbooks (e.g., What Is Mathematics? An Elementary Approach to Ideas and Methods) show that this is unaltered by projection. If C is between A and B the cross ratio is negative.

If this ratio is -1 then C and D are said to be 'Harmonic Conjugates' with respect to A & B
The following construction always produces the harmonic conjugate of C with respect to A and B for all positions of P and Q
As C tends towards the midpoint of AB, D tends to infinity and the line PR becomes parallel to AB.

Harmonic conjugates construction.

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