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Center of Gravity of Perimeter (Posted on 2008-05-18) Difficulty: 3 of 5
What is the center of gravity of the perimeter of a triangle (as when a piece of wire is bent into triangular form)?

  Submitted by Bractals    
Rating: 5.0000 (1 votes)
Solution: (Hide)
Let ABC be the triangle and A'B'C' its medial triangle.

Let the sides of triangle ABC be replaced with point masses (proportional to their length) located at the midpoints A', B', and C'.

If G is the center of gravity, then
          (ka)A'A' + (kb)A'B' + (kc)A'C'
   A'G = --------------------------------
                   ka + kb + kc

              1
       = ----------- [(b)A'B' + (c)A'C']
          a + b + c

              1                  A'B'                A'C'
       = ----------- [(b|A'B'|)-------- + (c|A'C'|)--------]
          a + b + c             |A'B'|              |A'C'|      

              1               A'B'             A'C'
       = ----------- [(bc/2)-------- + (cb/2)--------]
          a + b + c          |A'B'|           |A'C'|      

                bc         A'B'       A'C'
       = -------------- [-------- + --------]
          2(a + b + c)    |A'B'|     |A'C'|      
Clearly A'G bisects angle C'A'B'. A similar argument would show that B'G and C'G bisect angles A'B'C' and B'C'A' respectively.

Therefore, the center of gravity of the perimeter of a triangle is the incenter of its medial triangle.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some ThoughtsPuzzle ThoughtsK Sengupta2023-01-05 21:42:07
Solutionre(3): solutionCharlie2008-05-20 11:04:47
SolutionNew approachFrankM2008-05-20 09:18:57
Solutionre(2): solutionDej Mar2008-05-20 09:12:56
Some ThoughtsWhat's wrong with centroid invariance?FrankM2008-05-20 09:08:24
re(2): C of Gbrianjn2008-05-19 21:32:57
re: C of GCharlie2008-05-19 12:19:01
re: solutionCharlie2008-05-19 12:14:45
re(2): Centroid invariance in the face of increasing triangle sizeCharlie2008-05-19 12:05:01
re: Centroid invariance in the face of increasing triangle sizeCharlie2008-05-19 11:52:37
SolutionCentroid invariance in the face of increasing triangle sizeFrankM2008-05-19 11:19:49
C of Gbrianjn2008-05-19 10:27:52
solutionDej Mar2008-05-19 06:08:00
Solutionsimple but not simpleCharlie2008-05-18 18:02:26
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