Ponder This Challenge - October 2006 - Fraction of area of triangle

Ponder This Challenge:

Puzzle for October 2006.

Revised, 10/3/06, 11:30AM ET: Consider 3 points chosen at random in the interior of a triangle. We ask what is the expected (average) area of the triangle determined by these 3 points as a fraction of the area of the triangle they are chosen in. It is fairly easy to see that this fraction is independent of the size or shape of the original triangle. So we may assume we are choosing points in the triangle, S, in the x-y plane with vertices (0,0), (1,0) and (0,1). Let R be the triangle determined by the three random points. The value we are asking for can be computed by answering the following questions.

1. Consider the minimum triangle (with edges parallel to the edges of S), T, containing the 3 random points. What is the ratio of the expected area of T to the area of S.

2. There are 2 configurations of T and the 3 random points which occur with positive probability. These are:

• Case A - One random point is a vertex of T, another lies along the opposite side and the third is in the interior of T.* Case B - Each edge of T contains one of the 3 random points.

What is the probability of occurrence of each of these cases?

3. What is the ratio of the expected area of R to the area of T for each of the cases A and B above?

4. Combining the answers to questions 1-3, what is the ratio of the expected area of a triangle, R, formed by choosing 3 points at random in another triangle, S, to the area of S.

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Solution

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  Answer:

  We give the answers to each part with a sketch of the proofs.

  1. 5/14. It is easy to see that T is formed by removing a strip along each side of S and that each strip has expected area 1/4 of the area of S. However we have to add back the area near the corners of S which lies in two strips and is being counted twice. For the corner near 0 this area has expected value

  Integral from 0 to 1 with respect to x (Integral from 0 to 1-x with respect to y ((1-x-y)**6)) = 1/56.

  Dividing by the area of S this gives a ratio of 1/28. It can be shown the result for the other corners is the same. So the expected area of T is (1-3*1/4+3*1/28) = 5/14 of the area of S.

  2. 3/5, 2/5. Let the sides of T have lengths a,b,c where a is the length of the vertical side, b is the length of the horizontal side and c is the length of the diagonal side. Let the sides have infinitesimal widths da,db,dc respectively. Then the hypervolume of a type A configuration with corner near 0 is (da*db)*(c*dc)*(.5*a*b)=.5*a*b*c*da*db*dc. Similarly the other two type A configurations have the same hypervolume. For example (da*dc/sin(r))*(b*db)*(.5*a*c*sin(r)) =.5*a*b*c*da*db*dc (where r is the angle between the vertical and diagonal sides of T) for the type A configurations with top corner. On the other hand type B configurations have hypervolume (a*da)*(b*db)*(c*db) = a*b*c*da*db*dc. So in total the type A configurations have hypervolume 3/2 the hypervolume of the type B configurations. The result follows.

  3. 2/9, 1/4. For type A the corner point and the point on the opposite side divide T into 2 triangles. The third point lies in one of these triangles and divides it into 3 triangles. On average the third point lies in a triangle with 2/3 the area of T (since it is more likely to lie in big subtriangle) and divides it into equal parts. So the desired expected area ratio is (2/3)*(1/3)=2/9. For type B the expected area is what is left over when the corners of T are removed. Each corner has expected area 1/4 the area of T leaving expected area 1/4 of the area of T for the triangle determined by the three points.

  4. 1/12. The calculation is (5/14)*((3/5)*(2/9)+(2/5)*(1/4)) = (5/14)*((2/15) + (1/10)) = (5/14)*(7/30) = 1/12.

  This problem was asked by Watson (1865) and solved by Sylvester. I came up with the above derivation myself but it seems doubtful it is original.

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  If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solvers

• V. Balakrishnan (10.02.2006 @06:36:22 PM EDT)
• Alan Murray (10.03.2006 @01:49:54 AM EDT)
• Li Han (10.05.2006 @05:28:22 AM EDT)
• Roberto Tauraso (10.05.2006 @06:15:09 AM EDT)
• Arthur Breitman (10.05.2006 @10:40:20 AM EDT)
• Yurun Liu (10.06.2006 @12:52:45 PM EDT)
• Dan Dima (10.06.2006 @06:46:02 PM EDT)
• Zhou Guang (10.07.2006 @10:37:03 PM EDT)
• James Dow Allen (10.08.2006 @01:19:04 AM EDT)
• Phil Muhm (10.11.2006 @09:42:18 AM EDT)
• Yoav Raz (10.11.2006 @01:30:38 PM EDT)
• Dion K. Harmon (10.12.2006 @01:43:31 AM EDT)
• David McQuillan (10.13.2006 @10:26:42 AM EDT)
• Ashish Srivastava (10.16.2006 @03:59:09 AM EDT)
• Joaquim Neves Carrapa (10.16.2006 @04:30:11 AM EDT)
• Nyles Heise (10.18.2006 @03:42:38 AM EDT)
• John T. Robinson (10.18.2006 @03:51:25 PM EDT)
• Arjun Acharya (10.19.2006 @06:22:14 PM EDT)
• John Ritson (10.20.2006 @12:10:09 AM EDT)
• Se Kwon Kim (10.22.2006 @12:40:14 PM EDT)
• John G. Fletcher (10.26.2006 @01:27:24 PM EDT)
• Ray Gregory (10.31.2006 @12:33:29 AM EDT)