Mathematics › Properties of Triangles › Napiers Analogy, Area of Triangle & Half angle Formulae
With usual notations, in $\Delta \mathrm{ABC}$, if $\mathrm{b} \cos ^{2} \frac{\mathrm{C}}{2}+\mathrm{c}…
With usual notations, in $\Delta \mathrm{ABC}$, if $\mathrm{b} \cos ^{2} \frac{\mathrm{C}}{2}+\mathrm{c} \cos ^{2} \frac{\mathrm{B}}{2}=\frac{3 \mathrm{a}}{2}$, then
$\mathrm{b}, \mathrm{a}, \mathrm{c}$ are in A.P. $\mathrm{b}, \mathrm{a}, \mathrm{c}$ are in G.P. $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in G.P. $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in A.P. Solution
Given $\mathrm{b} \cos ^{2} \frac{\mathrm{C}}{2}+\mathrm{c} \cos ^{2} \frac{\mathrm{B}}{2}=\frac{3 \mathrm{a}}{2}$
$\therefore \quad \mathrm{b}\left(\frac{1+\cos \mathrm{C}}{2}\right)+\mathrm{c}\left(\frac{1+\cos \mathrm{B}}{2}\right)=\frac{3 \mathrm{a}}{2}$
$\therefore \mathrm{b}+\mathrm{b} \cos \mathrm{C}+\mathrm{c}+\mathrm{C} \cos \mathrm{B}=3 \mathrm{a} \Rightarrow(\mathrm{b} \cos \mathrm{C}+\mathrm{c} \cos \mathrm{B})+\mathrm{b}+\mathrm{c}=3 \mathrm{a}$
$\quad \mathrm{a}+\mathrm{b}+\mathrm{c}=3 \mathrm{a} \Rightarrow \mathrm{b}+\mathrm{c}=2 \mathrm{a} \Rightarrow \mathrm{a}=\frac{\mathrm{b}+\mathrm{c}}{2}$
$\therefore \quad \mathrm{b}, \mathrm{a}, \mathrm{c}$ are in A.P.
Asked in: MHT CET 2020 (13 Oct Shift 1)
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