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 Circumcircle and Incircle (Posted on 2009-05-26)
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 | Comment 3 of 4 |
`                                 inradius       rLet x = ratio of similitude = -------------- = ---                               circumradius     R`
`          area(ABC)        area(ABC)         1Then 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 anequilateral triangle.`
` `

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

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