If $8 R^{2} = a^{2} + b^{2} + c^{2}$ , then prove that the $△$ is right angled.

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Solution

$a^{2} + b^{2} + c^{2} = 8 R^{2}$
${sin}^{2} A + {sin}^{2} B + {sin}^{2} C = 2$
$3 - (cos 2 A + cos 2 B + cos 2 C) = 4$
$cos 2 A + cos 2 B + cos 2 C = - 1$
$- 1 - 4 cos A cos B cos C = - 1$
$cos A cos B cos C = 0 ⟹ cos A = 0$ or $cos B = 0$ or $cos C = 0$
$⟹ A = 90^{\circ}$ or $B = 90^{\circ}$ or $C = 90^{\circ}$
$⟹$ the triangle is right angled

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