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I was droodling a bit and a given moment I drew the following construction:

alt text

It appears that the three blue intersections are collinear (red line), no matter how I draw the construction lines. If this is always true, I assume that this a know fact [otherwise I have my first theorem! :-) erm conjecture, since I can't prove it :-( ].
What's the theorem called?
TIA
Steven

stevenvh
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1 Answers1

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Congratulations! Looks to me like you have rediscovered Pappus' Hexagon Theorem.

The image from the link:

alt text

Aryabhata
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  • That's it, thanks. Problem is that you can only google it if you know what it's called! – stevenvh Sep 03 '10 at 08:00
  • Looks like AC and DF don't even have to be parallel. – stevenvh Sep 03 '10 at 08:01
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    FYI, this works for 6 points on a conic section (Pascal's theorem). The case you have discovered is a degenerate case. –  Sep 03 '10 at 08:49
  • There are many such "configuration" theorems in projective geometry among which the other very famous theorem is due to Desargues:

    http://en.wikipedia.org/wiki/Desargues%27_theorem

     – Joseph Malkevitch Sep 03 '10 at 11:37
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    @steven: I googled it! "collinear points theorems" and I got this page: http://www.gogeometry.com/geometry/collinear_points_theorems_problems_index.htm – Aryabhata Sep 03 '10 at 13:47
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    @moron: No I hadn't tried it first, but at least it looks like I'm pretty good at choosing a title for my question! :-) Thanks for the link. +1 – stevenvh Sep 03 '10 at 14:39
  • @Stevenh: if you google "theorem three points collinear" the 2nd match is the Wikipedia page on Desargues theorem, which mentions Pappus's theorem in the 2nd paragraph. So it's really not that hard to find. – Bill Dubuque Sep 03 '10 at 16:04