There is an equilateral triangle of side length . It is inscribed in a circle, and a circle is inscribed inside the triangle. The ratio of the area of the incircle to the area of the circumcircle is?
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We know that Distance between incentre and cicumcentre = d = R ( R − 2 r ) where R and r are the Circumradius and Inradius respectively. (Thanks , Euler!)
R ( R − 2 r ) = 0 as the incenter and cicumcenter coincide for an equilateral triangle.
Now since R = 0 , ( R − 2 r ) has to be 0, for the distance to be 0.
Thus, R = 2 r
Ratio of areas= π ( 2 r ) 2 π r 2 = 4 1 = 0 . 2 5