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Geometry Level 5

( a b + c + 2 a ) ( b c + a + 2 b ) ( c a + b + 2 c ) 3 \large{\sqrt [ 3 ]{ \left( \frac { a }{ b+c+2a } \right) \left( \frac { b }{ c+a+2b } \right) \left( \frac { c }{ a+b+2c } \right) }}

Let a , b , c a,b,c be the sides of triangle A B C ABC with circumradius 2 and inradius 1. If the minimum value of the expression above is α β \frac { \alpha }{ \beta } for positive coprime integers α \alpha and β \beta , find α + β \alpha +\beta .


The answer is 5.

Hamza A by Hamza A , 19, USA

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1 solution

By Euler formula, square of distance between circumcenter and incenter is given by R 2 − 2 R r R^2-2Rr .In the given question evaluating OI we get OI=0.Hence triangle ABC is equilateral.Hence just put a=b=c in the expression to get the answer as 1/4.

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