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In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.

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#### Solution

Let the side of the equilateral triangle be *a*, and AE be the altitude of ΔABC.

`:. BE = EC = (BC)/2 = a/2`

Applying Pythagoras theorem in ΔABE, we obtain

AB^{2} = AE^{2} + BE^{2}

`a^2 = AE^2 + (a/2)^2`

`AE^2 = a^2 - a^2/4`

`AE^2 = (3a^2)/4`

4AE^{2} = 3*a*^{2}

⇒ 4 × (Square of altitude) = 3 × (Square of one side)

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