Name: Conditional Identities
Uploaded: 2019-07-19
Duration: 6 min 15 s
Description: The identities hold for all values of the angles that satisfy the given conditions among them are called conditional trigonometric identities. Watch this vid...

Question

In an equilateral triangle with usual notations the value of

\frac{27 r^{2} R}{r_{1} r_{2} r_{3}}

is equal to

Accepted Answer

$R$ = Radius of the circumference

$r$ = Radius of incentre

$r_{1}, r_{2}, r_{3}$ = Radius of the external circle at the vertex

$∴$ $R = \frac{a b c}{4 Δ}$ and $r = \frac{Δ}{s}$ where, $s = \frac{a + b + c}{2} = \frac{3 k}{2}$

Also, $r_{1} = \frac{Δ}{s - a}$ , $r_{2} = \frac{Δ}{s - b}$ and $r_{3} = \frac{Δ}{s - c}$

By Heron's formula, $Δ = \sqrt{s (s - a) (s - b) (s - c)}$

On squaring, $\frac{Δ^{2}}{s} = (s - a) (s - b) (s - c)$

$∴ \frac{27 r^{2} R}{r_{1} r_{2} r_{3}} = \frac{27 \times {(\frac{Δ}{s})}^{2} \times (\frac{a b c}{4 Δ})}{(\frac{Δ}{s - a}) \times (\frac{Δ}{s - b}) \times (\frac{Δ}{s - c})}$

$\Rightarrow \frac{27}{4} \times {(\frac{Δ}{s})}^{2} \times (\frac{a b c}{Δ^{3}}) \times (s - a) \times (s - b) \times (s - c)$

$\Rightarrow \frac{27}{4} \times {(\frac{Δ}{s})}^{2} \times (\frac{a b c}{Δ^{4}}) \times \frac{Δ^{2}}{s}$

$\Rightarrow \frac{27}{4} \times \frac{a b c}{s^{3}}$

Given triangle is equilateral,

$∴ \frac{27 r^{2} R}{r_{1} r_{2} r_{3}} = \frac{27}{4} \times \frac{k^{3}}{27 k^{3}} \times 8 = 2$

In an equilateral triangle with usual notations the value of $\frac{27 r^{2} R}{r_{1} r_{2} r_{3}}$ is equal to

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Learn and Prepare for any exam you want !

In an equilateral triangle with usual notations the value of $\frac{27 r^{2} R}{r_{1} r_{2} r_{3}}$ is equal to