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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?

 Submitted by Old Original Oskar! Rating: 3.8000 (5 votes) Solution: (Hide) Pick any P. Draw lines from P to each vertex. These lines divide the equilateral triangle into three triangles. Each of these triangles has as its base one side of the equilateral triangle and as its height the perpendicular distance from P to that side. The sum of the areas of these triangles equals the area of the original triangle. So, the sum of the three distances from P to the sides, is the same as the height of the original triangle. Thus, the average distance to the sides is √3/6 times the side of the triangle.

 Subject Author Date Quick and dirty solution; not a proof Larry 2020-04-23 06:48:48 Answer K Sengupta 2008-10-03 02:42:58 Any triangle Federico Kereki 2005-11-24 11:00:29 Solution Bractals 2005-11-24 10:41:53 Old problem goFish 2005-11-24 10:06:11 Solution Ken Haley 2005-11-24 09:44:07

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