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.

Posted by Bractals
on 20090526 13:36:54 