Medians, midlines, centroids and circumcenters | Universal Hyperbolic Geometry 17 | NJ Wildberger

Here we introduce basic aspects of triangle geometry into the superior framework of universal hyperbolic geometry, a purely algebraic setting valid over the rational numbers. We begin by reviewing the centroid and circumcenter in the Euclidean setting. In the hyperbolic plane, midpoints of a side don't always exist. If we consider a triangle in which each side has midpoints, there are then 6 medians, and their dual lines, called midlines here, although they play the role of perpendicular bisectors. The medians meet in 4 centroids. The midlines meet in 4 circumcenters.

There are some remarkable connections between centroids and circumcenters, culminating in the z point of the triangle. Remarkably it lies on the ortho-axis, and together with the base center, orthocenter and orthostar, forms a harmonic range of points. THANKS to EmptySpaceEnterprise for help with Video Contents.

Video Content:
00:00 Introduction
04:25 Euler line
06:06 Midlines of a side
9:47 # of midpoints of a triangle; duals of midpoints
18:27 Meets of medians theorem
20:14 construction of meets of midlines (circumcenters)
25:01 Centroid circumcenter correspondence theorem
29:06 Z-point ortho-axis theorem

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[njwildberger]

Its not so bad: over the rational numbers we can often approximately find these midpoints: if 1-q(a, b) is a square, then midpoints of the side ab will exist as it turns out. So if 1-q(a, b) is positive, it is approximately a square, and so approximate midpoints exist. You have to get used to this more precise description of what is happening, but that is nature!

The dynamic geometry packages generally work only to approximate decimal values.

[Cor]

Well, actually my observation is somewhat trivial. We actually have constructed constructed the z-lines to meet at those g and c points :)