I am struck at the last step. Please help me to proceed further.
2026-03-25 15:17:56.1774451876
Greatest value of (cot(A/2)+cot(B/2)+cot(C/2))/(cotA+cotB+cotC) in triangleABC
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Cauchy-Schwarz, or just $(a-b)^2\ge0$, implies $$ a^2+b^2\ge 2ab,$$ and so on for $b,c$ and $c,a$. Add them all up we got $$2(a^2+b^2+c^2) \ge 2(ab+bc+ca),$$ which means $$ \frac{ab+bc+ca}{a^2+b^2+c^2}\le 1.$$ That would solve your last step.
And Xander was right, try to post in LaTeX markup.