Apollonius and Harmonic Conjugates - Universal Hyperbolic Geometry 2 - NJ Wildberger

Explore the concept of harmonic conjugates in Universal Hyperbolic Geometry through this 38-minute lecture. Delve into Apollonius' introduction of harmonic conjugates and their relationship to four points on a line. Discover how pole polar duality associated with a circle produces families of harmonic ranges. Examine the occurrence of harmonic ranges in vertex bisectors, vector combinations, and quadrangle sides. Learn about polarity in general conics, find the polar of a null point, and understand various types of geometry. Investigate examples of harmonic ranges, harmonic pencils, and related theorems. Gain insights into harmonic vector combinations and the Harmonic Quadrangle Theorem.

Introduction
General conics
Harmonic conjugates
Displacements
Harmonic Ranges
Harmonic Pencil
Harmonic Quadrangle Theorem