Triangle Geometry Old and New - An Introduction to Hyperbolic Triangle Geometry

Explore an introduction to hyperbolic triangle geometry in this lecture from the Insights into Mathematics series. Discover classical results in Euclidean triangle geometry before delving into the universal hyperbolic geometry framework using a projective Beltrami Klein model. Learn about perpendicularity, midpoints, and bilines through simple projective constructions. Examine fascinating concepts like orthocenters, the Double triangle, Apollonian points, incircles, circumcenters, medians, and centroids. Gain insights into both classical and universal hyperbolic geometry through visual illustrations without complex formulas, making the content accessible to high school students and mathematics enthusiasts alike.

Introduction
Special Points
Circumcenter
The Simpson Line
The Hypocycloid
Incenters
firebox theorem
Gurgaon points
Isaw agonal conjugates
Isotonic conjugate
Amateur investigation
Necklaces
Hyperbolic Geometry
Universal Hyperbolic Geometry
Simple Hyperbolic Geometry
Associated Lines
A is Outside
Duality
Altitude
Point perpendicular to itself
Introducing a triangle
Introducing the orthocenter
Introducing the orthoaxis
Arcs Theorem
Parallelism
Theorem
Perspective
Or Thick Triangle
Ortho Axis
Midpoints and Bylines
Apollonian Points
Apollonian Circles
In Circles
Contact Points
Midpoints
Circum circles
Centroids
Theorems
References