Explore the fundamental concepts of triangle geometry within the framework of universal hyperbolic geometry in this 34-minute video lecture. Delve into the algebraic setting of rational numbers as you learn about medians, midlines, centroids, and circumcenters. Discover how midpoints of triangle sides behave differently in hyperbolic geometry, and examine the relationships between 6 medians and their dual lines. Investigate the fascinating connections between 4 centroids and 4 circumcenters, culminating in the z-point theorem. Gain insights into the Euler line, construction techniques, and the remarkable harmonic range formed by key triangle points on the ortho-axis.