 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Triangle Donut (Posted on 2006-01-01) 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) Comments: ( Back to comment list | You must be logged in to post comments.) 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 Please log in:

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