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T-N-T (Posted on 2017-11-25) Difficulty: 3 of 5

  
Let ABC be an arbitrary triangle with points D, E, and F
lying on rays AB, BC, and CA respectively such that

   AD/AB = BE/BC = CF/CA = x.

Prove that Area(ΔDEF)/Area(ΔABC) = 3x2 - 3x + 1.
  

  Submitted by Bractals    
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Solution: (Hide)

  
   Area(ΔDEF) = ½| FD ⊗ FE | =

   ½| [xAB + (x - 1)AC] [(1 - x)AB + (2x - 1)AC] | =

   ½ [x(2x - 1) - (x - 1)(1 - x)] | AB ⊗ AC | =

   [ 3x2 - 3x + 1 ] * Area(ΔABC)

QED
  

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Questionre: SolutionBractals2017-11-28 11:25:08
SolutionSolutionJer2017-11-26 11:16:42
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