Given a triangle in 3d space ABC, we know it's normal is

n =(B-A)x(C-A)

We also know that a point r in the plane of the triangle satifies

n.(r - A) = 0

Since we know two of the coordinates of r, we can rearrange to calculate the missing third coordinate.

Expand the brackets

n.r - n.A = 0

Add n.A to both sides

n.r = n.A

Expand left hand dot product

nx*rx + ny*ry + nz*rz = n.A

And rearrange to get rz

nz*rz = n.A cm - nx*r - ny*ry

rz =  (n.A - nx*rx - ny*ry)/nz