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Triangle Donut (Posted on 2006-01-01) Difficulty: 2 of 5
If X is the area of any triangle-PQR and you make a hole in the center of this triangle by first joining the midpoints of each side to create smaller triangle-STU and then again joining the midpoints of triangle-STU to create the hole in the center of this triangle, it would create a donut something like this:
          Q 
         / \
        /   \
       /     \
    S /_______\ U
     / \  A  / \
    /   \/_\/   \
   /     \ /     \
P /_______V_______\ R
          T
What is the area of the triangle donut in terms of X?

See The Solution Submitted by tanx    
Rating: 3.0000 (1 votes)

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A footnote | Comment 8 of 14 |

One way to prove that the area does indeed quarter on each iteration is to look at the semi-perimeter formula for the area of a triangle

A = Sqrt( s (s-a) (s-b) (s-c) ) where s = (a+b+c)/2.

Halving the lengths of each of the sides, halves each of the four terms inside the Sqrt, (1/2)^4.  Taking the square root leaves (1/2)^2 = 1/4 A.


  Posted by goFish on 2006-01-02 12:03:34
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