Orthocenters exist! | Universal Hyperbolic Geometry 10 | NJ Wildberger
In classical hyperbolic geometry, orthocenters of triangles do not in general exist. Here in universal hyperbolic geometry, they do. This is a crucial building block for triangle geometry in this subject. The dual of an orthocenter is called an ortholine---also not seen in classical hyperbolic geometry! This lecture also introduces a number of basic important definitions: that of side, vertex, couple, triangle, trilateral. We also introduce Desargues theorem and use it to define the polar of a point with respect to a triangle. The lecture culminates in the definition of the orthic line, orthostar and ortho-axis of a triangle. The ortho-axis will prove to be the most important line in hyperbolic triangle geometry. Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary. Video Content 00:00 Introduction 3:47 Definitions: couple, triangle, trilateral 10:45 Altitude line theorem 12:46 Altitude point theorem 14:59 Orthocenter theorem 18:24 Proof of Orthocenter theorem 29:39 Ortholine theorem 32:26 Desargues' theorem 34:31 Cevians and Desargues' polars 39:05 Orthic axis, orthostar and ortho-axis ************ 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. Wild Egg Maths YT channel: https://www.youtube.com/channel/UCriF... 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) Wild Egg Maths Playlists: • 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)
