WT55: Chromogeometry and the Omega triangle

Chromogeometry allows us to study not just one, but three Euler lines of a triangle. One for each of the three fundamental planar geometries: Euclidean (blue) and the two relativistic geometries called red and green. These Euler lines turn out to be nothing but the medians of the triangle of orthocenters of the original triangle, called the Omega triangle. Many interesting features arise relating orthocenters, circumcenters and nine-point centers of different colours.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.

Category:

Science & Technology

Tags:

• Geometry
• rational
• trigonometry
• chromogeometry
• Euler
• line
• Wildberger
• relativistic
• geometry
• Omega
• triangle
• orthocenter
• circumcenter
• nine-point
• center

[benthurston27]

I really think Euler himself would be pleased with this result. I might have to get Geometer's sketchpad and check for myself, but do you think the Omega Triangle of the first Omega Triangle is the original triangle, or yet another triangle? In which case I wonder if you ever do get back to the original by taking successive Omega Triangles?

[njwildberger]

@benthurston27 You can also get free software that does more or less the same things as Sketchpad, namely C.a.R. (by Rene Grossman) and Geogebra.

As for iterating the Omega triangle, that is an obviously interesting idea, and I have experimented somewhat with that, but could not find any periodicity.

[RSKueffner]

I presume you are familiar with Sir William Rowan Hamilton's articles on anharmonics.

[njwildberger]

No I am not. Please explain what it is roughly and why it might be relevant, thanks.

[benthurston27]

@benthurston27 It looks plausible that the blue orthocenter of the Omega Triangle is A2. And if I'm doing it right I think A1 might be the red orthocenter of the Omega Triangle and A3 the green but a couple times it only looked to me like the colored orthocenters from two of the vertices intersected at a point and not all 3 but I know they are all three supposed to meet at the same point, so I don't know.

[ilovehb94]

lol nice drum stick but very helpfull as i am curently doing an assignment at school

thankyou