WildTrig9: The laws of proportion for a triangle

Using rational trigonometry we develop Menelaus' and Ceva's theorem and some related results, namely the Law of Proportions, and the Alternating spreads theorem.

Category:

Science & Technology

Tags:

• Rational
• trigonometry
• Menelaus
• Ceva
• Alternating
• spreads
• theorem
• laws
• of
• proportion

[relike868p]

I recheck what the term refers to on Wikipedia, and I suppose it means the duality in the projective geometry setting.

[njwildberger]

@relike868p There are two somewhat different meanings of duality in projective geometry: one refers to the general similarity in the subject between points and lines, the other to a specific pairing between points and lines induced by a conic.

[relike868p]

I have seen from the Wikipedia article that Menelaus' theorem is dual to Ceva's theorem, but I just cannot see why. Can you explain it?

[njwildberger]

@relike868p I am not sure what is meant by dual in this context: it is a word that has different meanings in particular contexts. Can you explain what is the question?

[benthurston27]

I have a theory that Menelaus was the name of the Minotaur in that labyrinth, he wouldn't actually eat you, just tie you up and lecture you with geometry until you died of starvation/boredom.

[upton9265]

Apologies - should have waited until I saw the entire video! You noted the correction. Thanx again!

[upton9265]

In your Ceva's theorem, the last P3 should be an R3. FWIW. Great stuff, Norman!

[upton9265]

Good stuff! The whole series is great. I love your book as well. I'd be interested if anyone has applied the formulas to do computational geometry, i.e., determining intersections of lines and objects, voronoi tessleations, etc..