In a equilateral $△ A B C, A D ⊥ B C$ , prove that $A D^{2} = 3 B D^{2}$

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Solution

In Equlateral $Δ A B C$
$A D = \frac{\sqrt{3}}{2} a$
$B D = \frac{a}{2}$
$A D^{2} = 3 \frac{a^{2}}{4} 3 {(\frac{a}{2})}^{2} = 3 B D^{2}$

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△ A B C

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