WildTrig23: Highlights from triangle geometry (II)

Equilateral triangles, Napoleon's theorem and Morley's theorem are discussed from the view of rational trigonometry. We give an example over a finite field with 11 elements.

Category:

Science & Technology

Tags:

• Rational
• trigonometry
• equilateral
• triangles
• Napoleon's
• theorem
• Morley's
• finite
• fields

[benthurston27]

About 7:30; In 29 you show how to trisect a segment, if you trisected the segment opposite the angle and draw from those points to the vertex wouldn't that split the angle into 3 equal pieces? Maybe the construction you show in 29 for trisecting a segment can't be done with just a compass and straightedge?

[njwildberger]

@benthurston27 No, trisecting a segment and joining those points to the opposite vertex does not trisect the angle. Try this yourself with an actual diagram to convince yourself. However the trisection of a segment definitely is possible with compass and straightedge.

[benthurston27]

@njwildberger Ah, in my drawings it keeps looking like it breaks the angle down into 1:1.5:2 proportion. Thanks. One other thing I was thinking is I think I remember reading that Archimedes knew you could trisect the angle if you have a ruler instead of a straightedge, but I can't remember if it was his decision to eschew that kind of geometry with a metric for Euclid's type geometry or someone later. Anyway, it seems to me your desire is to reverse that decision. :)