Question

In any triangle ABC, a square+b square+c square divided by R square has the maximum value of​

Answer 1

Answer:

$\bf\frac{a^2+b^2+c^2}{R^2}\leq12$

Step by step explanation:

$\frac{a^2+b^2+c^2}{R^2}$

Using Sine formula:

$\boxed{In\:\triangle\:ABC, \frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}=2R}$

$=\frac{(2R\:sinA)^2+(2R\:sinB)^2+(2R\:sinC)^2}{R^2}$

$=\frac{(4R^2sin^2A)+(4R^2sin^2B)+(4R^2sin^2C)}{R^2}$

$=\frac{4R^2[sin^2A+sin^2B+sin^2C]}{R^2}$

$=4[sin^2A+sin^2B+sin^2C]$

using

$\boxed{\text{For all A, }-1\leq\:sinA\:\leq\:1}$

$\leq4[1+1+1]$

$=12$

$\implies\boxed{\bf\frac{a^2+b^2+c^2}{R^2}\leq12}$