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 Perpendiculars Picking Probability (Posted on 2005-11-24)
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: 3.8000 (5 votes)

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 Solution | Comment 3 of 6 |
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`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.`
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 Posted by Bractals on 2005-11-24 10:41:53

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