How to evenly disperse sphere's on a 3 Dimensional k-Sphereflake (Kochs Snowflake)

Question

I am trying to find a way to disperse 9 sphere's evenly atop another sphere, essentially I want it to look like something in the pictures below. The first 6 spheres I have calculated for, I have evenly dispersed them in a hexagonal form along the equator of the sphere.Then I have 3 remaining, which are suspended in 3-dimensional space on the surface of the parent sphere (K1) they should also form an equilateral triangle, where each sphere also forms an equilateral triangle with the nearest spheres in the hexagonal ring.

Here you can see a sphere K1 with its parent sphere K0

Here is the hexagonal ring of spheres along the equator, with some triangle circumcisions to get its coordinates

There are two more images in the comments

Basically I am unsure how to obtain the coordinates of the middle points of the three spheres not in the hexagonal ring, as I am not the best at geometry let alone geometry in 3-Dimensional space.

Also to note: The origin of the coordinate system is the middle point of K1 and the y-axis intersects both K1 and K0 centre points