This inequality holds for any triangle:
where , and correspond to the lengths of the triangle.
Can you prove it?
Previous iteration: #760106 (brief and corrected answer in #760229; I will write out a more detailed version of this same answer as well as an alternative approach using the tau function or divisor function, when I am a bit less tired)
First, we replace with being tangents on the inscribed circle, as in
The inequality thus becomes
which is
or
First, for the upper bound
which is obviously true because is
Second, for the lower bound
Is equivalent to
The sum must be >= 0 because each summand is.
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The inequality with 1/3 is a special case of the Cauchy inequality 😉
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Here's proof
"Answer to the Ultimate Question of Life, the Universe, and Everything"
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