You choose a random point, uniformly, within an equilateral triangle.
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.

Posted by Bractals
on 20051124 10:41:53 