tan(A)tan(B)tan(C)

Geometry Level 4

Assuming A,B,C are all angles of a convex triangle, find the minimum of

t a n ( A ) + t a n ( B ) + t a n ( C ) . tan(A) + tan(B) + tan(C).

3sqrt(3) 2sqrt(3) 3sqrt(3)/2 sqrt(3)/2

Max Shav by Max Shav , 25, USA

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1 solution

Deeparaj Bhat
Feb 27, 2016

Put Z=tanA+tanB+tanC=tanAtanBtanC.

Now, by AM-GM, Z/3>=Z^(1/3).

Therefore, Z>=3√3.

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