Find the coordinate of third point of equilateral triangle. I have two points A and B whose coordinates are $(3,4)$ and $(-2,3)$ The third point is C. We need to calculate its coordinates.
I think there will be two possible answers, as the point C could be on the either side of line joining A and B.
Now I put AB = AC = BC.
We calculate AB by distance formula : $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
= $ \sqrt{5^2 + 1^2} = \sqrt{26}$
I plug this value into the distance of AC and BC and finally equate to get :
$5x + y = 6$
Now what can I do? There are two variables, I am getting equation of a line! How can I solve this?

If you like vector approach:
Displace (shift/translate) point A by vector(-3,-4) to come to the origin.
Multiply the new radius vector  $ab$  with $ e^ {i \pi/3} , e^ {-i \pi/3}$ (once clockwise and once anticlockwise) to obtain new points $ C_1 $ and $ C_2 $. The multiplying factor is $ (1/2 \pm i \sqrt 3/2) $.
Displace these points back to original positions translating by $ (3,4) $. 
If you multiply thrice, all points of a hexagon would also be reached.
