Of A(x1,y1) , B(x2,y2), C(x3,y3) are the vertices of an equilateral triangle whose side is equal to a , then prove that
|x1 y1 2 |^2
|x2 y2 2| = 3a^4
|x3 y3 2|

Dear student
Consider x1y12x2y22x3y32=x1y11+1x2y21+1x3y31+1=x1y11x2y21x3y31+x1y11x2y21x3y31   As if each element of a row column of a detreminant is expressed as a sum of two or more terms  then the determinant can be expressed as the sum of two or more determinants.Now we know that 


So, x1y11x2y21x3y31+x1y11x2y21x3y31=234a2+234a2=32a2+32a2=3a2So, x1y12x2y22x3y32=3a2Squaring both sides , we get x1y12x2y22x3y322=3a4
Regards

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