通过

Puzzle: geometry problem

  • 项目

Here is a fun one: The triangle ABC is isoseceles, with angle(ABC) = angle(ACB) = 80°. We draw now two segments inside the triangle. First, we choose the point D on segment AC such that angle(CBD) is 60°. Also, we choose the point E on AB, where angle(BCE) is 50°. We now extend DE until it intersects with the line on the BC segment in point F, as below:

The question is: what is the angle between the two lines on F? In other words, what is the value of angle(DFC)?

Comments

  • Anonymous
    April 11, 2005
    30 degrees.
  • Anonymous
    April 11, 2005
    30 deg
  • Anonymous
    April 11, 2005
    Would someone post how they got the solution at some point? I, with my hate of geometry, plugged at this problem and came up short. But I'd be anxious to see how the solution was gotten.

    Though, to be fair to other people who want to try on it before being given the solution, perhaps post the solution on another page and link to it, or obfuscate it in some other fashion.
  • Anonymous
    April 11, 2005
    The comment has been removed
  • Anonymous
    April 11, 2005
    Using the angles you have already worked out you now have one more isoceles triangle, can you see where?

    This means that to all intents and purposes you DO have a new length to play with (albeit a length relative to other known lengths).

    You can now use the full machinery of triangle geometry involving length! This means you can use the sine rule, similarity of triangles, Heron's formula (!!)...

    Some ways of finishing the problem off will be far more elegant than others, but for all intents and purposes noticing this reduces the problem to 'just' a computation.

    Slightly offtopic:

    For a really interesting tour of triangle geometry check out Paul Yiu's introduction here:
    http://www.math.fau.edu/yiu/TourOfTriangleGeometry/MAAFlorida37040428.pdf

    Also see Clark Kimberling's phenomenal compilation of triangle centers:
    http://faculty.evansville.edu/ck6/encyclopedia/ETC.html
  • Anonymous
    April 11, 2005
    hahaha!

    i forwarded this around to my office, and everyone is stuck working on it....at first it irritated me that i couldnt solve it, but now i take pleasure in knowing that the rest of them are struggling too...

    :P
  • Anonymous
    April 11, 2005
    You are an evil, evil man, Adi. Valorie and I have been on this WAY too long
  • Anonymous
    April 11, 2005
    BAC = 20
    ABD = 20
    BEC = 50
    BED = 70
    EDB = 40
    DEB = 70
    AED = 60
    ADE = 100
    BEF = 60 (since FED = 180)
    so eventually got
    BFE = 20
  • Anonymous
    April 11, 2005
    The comment has been removed
  • Anonymous
    April 12, 2005
    The comment has been removed
  • Anonymous
    April 12, 2005
    I can get
    this, it will take some time...


    At work you feel rushed to finish...
    ...
  • Anonymous
    April 12, 2005
    spent an hour..seems like you're almost there but not there yet :( doesn't seem to be too difficult but tricky enough to eat your brains. just hope its not something too trivial :-S
  • Anonymous
    April 12, 2005
    The comment has been removed
  • Anonymous
    April 12, 2005
    Search on Google for "isosceles 80" and click on the first result.
  • Anonymous
    April 12, 2005
    Argh. Ok, so I couldn't just use my two trivial equations of triangle angle and straight lines :)

    Oh well! Thanks :) (I was hoping that SOCATOA didn't come into it, I always hated that stuff!)
  • Anonymous
    April 13, 2005
    40/3 deg
  • Anonymous
    April 14, 2005
    The comment has been removed
  • Anonymous
    April 20, 2008
    PingBack from http://carmen.freetvnewsworld.com/solvingtrianglesgeometry180.html