Prove that the points (2a,4a), (2a,6a) and (2a+√3a,5a) are vertices of an equilateral triangle.

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Let ABC be the equilateral triangle. with vertices A(2a,4a) ,B(2a,6a) ,C(2a+√3a,5a)
AB= √((2a−2a)2+(6a−4a)2) = √(02+2a)2 = 2a

BC= √((2a+√3a−2a)2+(5a−6a)2) = √(3a2+a2) = 2a

CA= √((2a+√3a−2a)2+(5a -4a)2) = √(√3a)2+a2) = 2a

AB = BC = CA
∴ the vertices are of an equilateral triangle.

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