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Question Number 169271 by Shrinava last updated on 27/Apr/22

Answered by mr W last updated on 27/Apr/22

a=r cot (B/2)+r cot (C/2)  (a/r)=cot (B/2)+cot (C/2)≥2(cot (B/2) cot (C/2))^(1/2)   ((a/r))^6 ≥2^6 ×(cot (B/2) cot (C/2))^3   Σ((a/r))^6 ≥2^6 ×Σ(cot (B/2) cot (C/2))^3                  ≥2^6 ×3×(cot (A/2) cot (B/2) cot (C/2))^2                  ≥2^6 ×3×3^3 =5184 ✓

$${a}={r}\:\mathrm{cot}\:\frac{{B}}{\mathrm{2}}+{r}\:\mathrm{cot}\:\frac{{C}}{\mathrm{2}} \\ $$$$\frac{{a}}{{r}}=\mathrm{cot}\:\frac{{B}}{\mathrm{2}}+\mathrm{cot}\:\frac{{C}}{\mathrm{2}}\geqslant\mathrm{2}\left(\mathrm{cot}\:\frac{{B}}{\mathrm{2}}\:\mathrm{cot}\:\frac{{C}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$$\left(\frac{{a}}{{r}}\right)^{\mathrm{6}} \geqslant\mathrm{2}^{\mathrm{6}} ×\left(\mathrm{cot}\:\frac{{B}}{\mathrm{2}}\:\mathrm{cot}\:\frac{{C}}{\mathrm{2}}\right)^{\mathrm{3}} \\ $$$$\Sigma\left(\frac{{a}}{{r}}\right)^{\mathrm{6}} \geqslant\mathrm{2}^{\mathrm{6}} ×\Sigma\left(\mathrm{cot}\:\frac{{B}}{\mathrm{2}}\:\mathrm{cot}\:\frac{{C}}{\mathrm{2}}\right)^{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\geqslant\mathrm{2}^{\mathrm{6}} ×\mathrm{3}×\left(\mathrm{cot}\:\frac{{A}}{\mathrm{2}}\:\mathrm{cot}\:\frac{{B}}{\mathrm{2}}\:\mathrm{cot}\:\frac{{C}}{\mathrm{2}}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\geqslant\mathrm{2}^{\mathrm{6}} ×\mathrm{3}×\mathrm{3}^{\mathrm{3}} =\mathrm{5184}\:\checkmark \\ $$

Commented by Shrinava last updated on 27/Apr/22

Thankyou dear sir perfect

$$\mathrm{Thankyou}\:\mathrm{dear}\:\mathrm{sir}\:\mathrm{perfect} \\ $$

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