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.