How to find third coordinate of an equilateral triangle

So im looking for a formula or method of finding the third coordinate of a triangle. I'm just using these numbers for an example: i have a line going from the origin (0,0) to the point (3,2). This line is the base of my equilateral triangle. I figure the length of the base turns out to be sqrt(13) which holds true for the other edges of the triangle and the slope of the base line is 2/3.

Well to find the next point in the triangle i draw a line from the midpoint of the base to this mystery coordinate. This line I figure to have a slope of -3/2 and the height of the right triangle that is formed is sqrt( sqrt(13) ^2 - ( sqrt(13) / 2) ^2 ).

...and this is where I am stuck. Ive been trying to figure out where to go next for the past two days I'm desperate now please help.

The midpoint is (3/2,1). Your line from the midpoint to the mystery point is

y = (-3/2) ( x - 3/2 ) + 1 ..............(1)

The squared distance from the point to (0,0) is

x^2 + y^2 = 13, or

x^2 + [(-3/2) ( x - 3/2 ) + 1]^2 = 13. .............(2)

The two solutions can be found from equations (1) and (2). They are approximately

(-0.23205,3.59808) and (3.23205,-1.59808).

i not redoing z math i trusting it to be sound i just going to try explaining why there is two answers first instead of looking at your point i going to have you look at a point on the x axis as well as your origin now this is two point of a equilateral triangle but i can make one above the axis one below now using this as example you can see that solving for your point gives you two points up to you to decide which you want to use