What's the average distance to the three sides?

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.