Trigonometry & Inverse Trigonometry
Trigonometry
star_batch_jee_advanced_2025
Grade 11
Question:
If $\Delta$ be area of incircle of a triangle $ABC$ and $A_1, A_2, A_3$ be the area of excircles then find the least value of $$\frac{A_1A_2A_3}{729A^3}$$
Step-by-Step Solution
Key Concept: Use AM-GM inequality on the semi-perimeter differences to establish the lower bound.
The expression $\frac{\Delta_1\Delta_2\Delta_3}{\Delta^3}$ equals $\frac{(r_1r_2r_3)^2}{r^6} = \left(\frac{x}{s-a} \times \frac{x}{s-b} \times \frac{x}{s-c}\right)^2$. By AM-GM inequality on $(s-a), (s-b), (s-c)$, the denominator $(s-a)(s-b)(s-c) \geq 27$ with minimum value $1$.
Correct Answer: 1