### Triangle Geometry Old and New: An introduction to Hyperbolic Triangle Geometry

We present a very brief survey of a few classical results in Euclidean triangle geometry, and then give an introduction to triangle geometry in the new setting of universal hyperbolic geometry. This occurs in a projective Beltrami Klein model involving both the inside and outside of a distinguished circle. This lecture assumes no familiarity with either classical or universal hyperbolic geometry, and introduces only the concepts of perpendicularity, midpoints and bilines (analogs of angle bisectors) via simple projective constructions using the fixed circle. With this framework we present a nice cross section of interesting results involving orthocenters and the orthoaxis, the Double triangle, the x,z,b,h and s points, Incenters, Apollonian points and Incircles, Circumcenters and Circumcircles, Medians, Centroids etc. A good high school student can easily appreciate the results, which are illustrated by lots of pictures---there are no formulas in this talk! This talk was presented to the AMSI Summer School at UNSW in January 2012 by N J Wildberger.

Length:
01:05:17