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