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Question
Prove that in an equilateral triangle, three times the square of a side is equal to four times the square of its altitudes.
Solution
Let ABC be an equilateral triangle and let `AD ⊥ BC`.
In Δ ADB and Δ ADC we have
`AB=AC`
`∠ B =∠ C`
And ` ∠ ADB = ∠ADC`
So , `BD =DC`
`⇒ BD =DC = 1/2 BC`
Since Δ ADB is a right triangle right-angled at D. So
`AB^2=AD^2+BD^2`
`AB^2=AD^2+(1/2BC)^2`
`AB^2=AD^2+(AB^2)/4`
`3/4 AB^2 =AD^2`
`3AB^2=4AD^2`
Hence proved.
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