Projective harmonic conjugate

$D$ is the harmonic conjugate of $C$ with respect to $A$ and $B$ .
$A, D, B, C$ form a harmonic range.
$KLMN$ is a complete quadrangle generating it.

In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following construction:

Given three collinear points

A, B, C

, let

L

be a point not lying on their join and let any line through

C

meet

LA, LB

at

M, N

respectively. If

AN

and

BM

meet at

K

, and

LK

meets

AB

at

D

, then

D

is called the harmonic conjugate of

C

with respect to

A

and

B

.^[1]

The point $D$ does not depend on what point $L$ is taken initially, nor upon what line through $C$ is used to find $M$ and $N$ . This fact follows from Desargues theorem.

In real projective geometry, harmonic conjugacy can also be defined in terms of the cross-ratio as $(A, B; C, D) = -1$ .

^ R. L. Goodstein & E. J. F. Primrose (1953) Axiomatic Projective Geometry, University College Leicester (publisher). This text follows synthetic geometry. Harmonic construction on page 11

[1] R. L. Goodstein & E. J. F. Primrose (1953) Axiomatic Projective Geometry, University College Leicester (publisher). This text follows synthetic geometry. Harmonic construction on page 11

[1]