Could someone please help find the third coordinate of an equilateral triangle?

Question

the first two coordinates are ( a , 0 ) and ( 2a+at² , 0 ) please show your working

Answer 1

let  A( a , 0)  , B( 2a+at^2 , 0)

let third coordinate is  C(x ,y)

 A ( a , 0 )  B(2a+at^2 , 0) =  ( x1 ,y1) ,( x2 ,y2)

the mid point AB is  = (x1+x2)/2 ,(y1+y2)/ 2

                                   =(a +2a +at^2)/2,  (0 + 0)/2

                                  =(3a +at^2)/2 , 0

A ( a , 0 ) ,C ( x , y )  =( x1 ,y1) ,( x2 ,y2)

the mid point Ac is  = (x1+x2)/2 ,(y1+y2)/ 2

                                  = (a +x)/2 ,( 0 +y )/2

comparing the mid point AB &AC

 (3a +at^2)/2 = (a +x) /2

cancel  common  terms

 3a +at^2  = a +x

subtract a from each side

3a -a +at^2 =x

2a +at^2 = x

   0 = (0 +y)/2

  0 = y

the third coordinate ( x , y ) = ( 2a+at^2 , 0)