AC Geometry DCB Curves17 | Three points on a circle and their poles (new theorems?) | Wild Egg maths

We look at possible analogs of the parabola theorems we were investigating in our last video with Steven Choi. This time we look at the case of the circle and three points on it, determining three dual or polar lines and corresponding three external points. This should be considered just as a rough first foray into this general line of investigation. A natural question arises: while we have looked at the Centroid, Orthocentre and Circumcentre of the three associated triangles -- what happens if we consider instead one of the hundreds/ thousands of other triangle centers found in Clark Kimberling's Encyclopedia of Triangle Centers? This is part of our effort to resuscitate classical algebraic geometry which is primarily concerned with the GEOMETRIC, not algebraic, properties of curves and surfaces. A much overlooked topic in the modern view, but of principle importance to 18th and 19th century geometers, and ripe for a reinvigoration, due to the power of dynamic geometry packages, and also powerful algebraic software that can be used to simplify the search for proofs. (The AI's are soon going to be telling us all about that!) CORRECTION: When talking about the Orthocentre triangle, I stated that the orientations were reversed. This is not correct --the orientation between it and the original triangle are the same. :) ****** Research Gate page: https://www.researchgate.net/profile/... Blog: http://njwildberger.com/ Online courses at openlearning.com (currently Algebraic Calculus One): https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! Patreon:   / njwildberger   Your support would be much appreciated. Insights into Mathematics YT channel:    / njwildberger   Wild Egg Maths Playlists: (please become a Member to access all of them)    • Intro to Algebraic Calculus with Box Arith...   (4 videos)    • Classical to Quantum (for Members)   (64 videos)    • Solving Polynomial Equations and the Geode...   (45 videos)    • De Casteljau Bezier curves and associated ...   (20 videos)    • Exploring q-series (for Members)   (8 videos)    • Six: A mathematical exploration   (9 videos)    • Algebraic Calculus One: a new foundation f...   (52 videos)    • Advice to prospective research mathematici...   (9 videos)    • The Hexagrammum Mysticum: a Geometric Gem ...   (14 videos)    • Algebraic Calculus Two   (8 videos)    • Special (Polynomial) Functions and Maxel N...   (25 videos)    • Dynamics on Graphs: The ADE Phenomenon & B...   (30 videos) Insights into Mathematics Playlists:    • The Algebra of Boole, Logic and Circuit An...   (31 videos)    • Box Arithmetic: a new framework for Mathem...   (18 videos)    • Hypergroups and Diffusion Symmetry: an int...   (6 videos)    • Rational Trigonometry for maths, physics a...   (4 videos)    • Sociology and Pure Maths   (44 videos)    • Old Babylonian mathematics and Plimpton 322   (8 videos)    • Math Foundations   (226 videos)    • Math Seminars N J Wildberger   (26 videos)    • Math History (ancient to modern)   (45 videos)    • Geometric Linear Algebra   (43 videos)    • Algebraic Topology   (40 videos)    • Universal Hyperbolic Geometry   (55 videos)    • Differential Geometry   (34 videos)    • Elementary Probability and Statistics   (8 videos)    • Math Terminology for Incoming Uni Students   (9 videos)    • Famous Math Problems   ( 46 videos)    • Elementary Mathematics Explained (K-6)   (40 videos)    • Ancient Mathematics and insights of Howard...   (7 videos)    • Wild West Banking: A mathematician goes We...   (7 videos)    • Playing Go: the ancient oriental board game   (19 videos)    • Maths and Music   (21 videos)    • Year 9 Mathematics (review fractions, deci...   (10 videos)    • Wild Trig: An introduction to Rational Tri...   (94 videos)