For sphere of radius 2 centered at origin, let A,B,C be 3 points all on the sphere, all in a horizontal plane in negative z territory with side lengths all 3.  Make the y-coordinate of point A be zero.

A: ( √3, 0, -1 )

B: ( (-√3)/2, 3/2, -1 )

C: ( (-√3)/2, -3/2, -1 )

Find point D, s.t. its y-coordinate is zero, above the plane of ABC.

And s.t. it is a distance of 3 from B and C; if its y value is 0 then it will be equidistant from B and C.

At this point, I used a program to test values for D varying its x coordinate from 0 to -2, keeping its y coordinate at 0, and calculating its z coordinate as

z = ± √(4 - x^2) from the equation for the sphere.

This found

D: ( -1.237179, 0, 1.5714286881557813 )

I found the distance from A to D to be 3.9279219885871046

Curiously, the z coordinate found for D is very close to 11/7