Projective harmonic conjugate

In <a href="/facts/Projective_geometry/rHKSaEbx">projective geometry</a>, the harmonic conjugate point of a point on the <a href="/facts/Real_projective_line/1g8z8v3U">real projective line</a> 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.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
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 <a href="/facts/Desargues_theorem/vJzx0uPh">Desargues theorem</a>.
In real projective geometry, harmonic conjugacy can also be defined in terms of the <a href="/facts/Cross-ratio/tj4DK3Vk">cross-ratio</a> as (A, B; C, D) = −1.

Projective harmonic conjugate open-in-new

Projective harmonic conjugate