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Circumcircle and Incircle (Posted on 2009-05-26) Difficulty: 2 of 5
ABC and XYZ are similar triangles and the circumcircle of the triangle XYZ is the incircle of the triangle ABC. If k = Area of ABC/Area of XYZ, then find the minimum value of k.

See The Solution Submitted by Praneeth    
Rating: 4.0000 (1 votes)

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Solution Solution | Comment 3 of 4 |

                                 inradius       r
Let x = ratio of similitude = -------------- = ---
                               circumradius     R
          area(ABC)        area(ABC)         1
Then k = ----------- = ----------------- = -----
          area(XYZ)     x^2 * area(ABC)     x^2
    r                                     1                              
   --- = cos(A) + cos(B) + cos(C) - 1 <= ---
    R                                     2
   with equality when A = B = C.
Therefore, the minimum k is 4 when ABC is an
equilateral triangle.

 

  Posted by Bractals on 2009-05-26 13:36:54
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