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Apollonius introduced the important idea of harmonic conjugates, concerning four points on a line. He showed that the pole polar duality associated with a circle produces a family of such harmonic ranges, one for every line through the pole of a line. Harmonic ranges also occur in the context of vertex bisectors, as combinations of vectors, and associated with the sides of a quadrangle. CONTENT SUMMARY: polarity holds for general conics @03:58 geometers sketchpad in use @05:02 How to find the polar of a null point @05:50 Harmonic conjugates @08:58 discussion of various types of geometry @12:17 More on harmonic conjugates @16:37 examples of harmonic ranges and harmonic ranges theorem @24:00 Harmonic pencils and Harmonic bisectors theorem @28:34 Harmonic vector combinations theorem @32:37 Harmonic quadrangle theorem @34:34 ************************ Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects! If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated. *********************** Here are all the Insights into Mathematics Playlists: Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist? list=PL8403C2F0C89B1333 Year 9 Maths: https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK Ancient Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e Wild West Banking: https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH Sociology and Pure Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist? list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov Math History: https://www.youtube.com/playlist?list=PL55C7C83781CF4316 Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list=PL3C58498718451C47 MathFoundations: https://www.youtube.com/playlist?list=PL5A714C94D40392AB Wild Linear Algebra: https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp Famous Math Problems: https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI Differential Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP Algebraic Topology: https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8 Math Seminars: https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF ************************ And here are the Wild Egg Maths Playlists: Triangle Centres: https://www.youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS Algebraic Calculus One: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB Algebraic Calculus Two: https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG- м
