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Triangle inscribed in a triangle with smallest perimeter. (Posted on 2010-04-25) Difficulty: 4 of 5
Given an acute triangle. Find an inscribed triangle whose perimeter is minimum.

See The Solution Submitted by Vee-Liem Veefessional    
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Solution | Comment 1 of 3

Let tPQR denote triangle PQR.
Let ABC be the acute triangle and A'B'C' 
the inscribed triangle with A', B', and
C' on sides opposite vertices A, B, and
C respectively.
For the perimeter of tA'B'C' to be a
minimum, then tA'B'C' must satisfy
  tAB'C' ~ tA'BC' ~ tA'B'C ~ tABC.
The orthic triangle of tABC (AA', BB',
and CC' are altitudes of tABC) is the
only triangle that does.

 


  Posted by Bractals on 2010-04-25 15:03:17
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