Harmonic Ratio in Complex Domain
A straight line
For any four points on such a line defined by four parameter values p, q, r, s, the crossratio is obtained easily in terms of the (real) parameter values:
(f(p), f(q); f(r),f(s)) = (p, q; r, s).
Let's verify that this is indeed so.
(1)  (f(p), f(q); f(r), f(s)) = (f(p)  f(r))/(f(p)  f(s)) : (f(q)  f(r))/(f(q)  f(s)). 
Evaluating a piece at a time,
f(p)  f(r)  = (a + pb)/(1 + p)  (a + rb)/(1 + r) 
= [(a + pb)(1 + r)  (a + rb)(1 + p)] / (1 + p)(1 + r)  
= (a + pb + ra + prb  a  rb  pa  prb) / (1 + p)(1 + r)  
= (pb + ra  rb  pa) / (1 + p)(1 + r)  
= (p  r)(b  a) / (1 + p)(1 + r), 
and similarly for the other three differences in (1). Further,
(f(p)  f(r))/(f(p)  f(s))  
= (p  r)(1 + s) / (p  s)(1 + r). 
Finally, for the crossratio, we see that
(f(p), f(q); f(r), f(s))  = (f(p)  f(r))/(f(p)  f(s)) : (f(q)  f(r))/(f(q)  f(s)) 
=  
= (p  r)(1 + s)(q  s)(1 + r) / (p  s)(1 + r)(q  r)(1 + s)  
= (p  r)(q  s) / (p  s)(q  r)  
= (p  r)/(p  s) : (q  r)/(q  s), 
as promised. We fix now points A(a) and B(b) and concentrate on the values p and q that correspond two points P(p) and Q(q) harmonically conjugate with respect to A and B:
(p, q; 0, ∞) = 1.
So,
1  = (p  r)/(p  s) : (q  r)/(q  s) 
= (p  r)/(q  r) : (p  s)/(q  s)  
= (p  0)/(q  0) : (p  ∞)/(q  ∞)  
= p/q, 
which implies q = p! If we define a conjugation function, say,
F(F(P))  = F(Q) 
= P. 
A function with this property is known as involution. We use this to establish two important properties of harmonic conjugation:
Proposition
Both are proved by direct verification. For example,
f(p)  a  = (a + pb)/(1 + p)  a 
= (a + pb  a  pa)/(1 + p)  
= p(b  a)/(1 + p). 
Similarly
f(q)  a  = q(b  a)/(1 + q) 
= p(b  a)/(1  p). 
Adding the two terms in #1, we obtain
1/(f(p)  a) + 1/(f(q)  a)  = (1 + p)/[p(b  a)]  (1  p)/[p(b  a)] 
= 2/(b  a). 
The second part of the proposition is as straightforward. Just note that f(1) = (a + b)/2 is the geometric image of the midpoint of the segment AB. A noteworthy fact about the proposition is that the right hand sides in both identities are independent of the selection of the pair of conjugates
which we use in one of the proofs of the Butterfly theorem.
References
 C. Zwikker, The Advanced Geometry of Plane Curves and Their Applications, Dover, 2005
Poles and Polars

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