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Perpendiculars Picking Probability (Posted on 2005-11-24) Difficulty: 3 of 5
You choose a random point, uniformly, within an equilateral triangle.

What's the average distance to the three sides?

See The Solution Submitted by Old Original Oskar!    
Rating: 3.5000 (4 votes)

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Solution Solution | Comment 3 of 5 |
 
Let ABC be the equilateral triangle with side length s and P the point.
Connect the point P to the vertices of ABC with line segments.
  Area(ABC) = Area(PAB) + Area(PBC) + Area(PCA)
  1                   1                   1                   1
  - s Altitude(ABC) = - s Altitude(PAB) + - s Altitude(PBC) + - s Altitude(PCA)
  2                   2                   2                   2
  Altitude(ABC) = Altitude(PAB) + Altitude(PBC) + Altitude(PCA)
  Altitude(PAB) + Altitude(PBC) + Altitude(PCA)   Altitude(ABC)    s sqrt(3)
  --------------------------------------------- = ------------- = -----------
                       3                                3              6
Note: If point P is not restricted to the interior of ABC, then
      Altitude(ABC) = pc * Altitude(PAB) + pa * Altitude(PBC) + pb * Altitude(PCA)
      where pa = 1 if P and A are on the same side of line BC; -1 otherwise.
            pb = 1 if P and B are on the same side of line CA; -1 otherwise.
            pc = 1 if P and C are on the same side of line AB; -1 otherwise.
 

  Posted by Bractals on 2005-11-24 10:41:53
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