What is the tangent plane equation on the 3 spheres?

First solve some quadratic equations to find $y_2$, $x_3$, and $y_3$.

Then, since each of the spheres is tangent to $z=0$ as well as to the mystery plane, its center must lie on the plane that bisects the angle made between these two planes. We can compute an equation for bisecting plane, because that is defined by the centers.

Now reflect the plane $z=0$ about the bisecting plane.

I get: -150.235 x-89.3299 y-128.83 z+1729.87=0 for the tangent plane equation. (Close approximation.)

Tangent points:

Sphere #1: {3.4594896600074643,2.05701926455194,7.966581035101458`}

Sphere #2: {2.0756918905582737,8.980177236680767,4.779953859745328`}

Sphere #3: {5.375454589335715,5.728584817849962,3.1866359029829208`}

Sphere center points:

Sphere #1 center point: {0, 0, 5}

Sphere #2 center point: {0., 7.74597, 3.}

Sphere #3 center point: {3.99166, 4.90578, 2.}

I calculated the tangent plane equation by using a plane thru 3 points. The first 2 points are apexes of 2 cones that envelope sphere #1 and sphere #2, then sphere #1 and sphere #3. Think of the line that connects the apexes as a hinge line that lies in the tangent plane. Select a third point whose x and y coordinates will be close to the tangent point on sphere #1. Reduce or add to the z coordinate as required to lower or raise the tangent plane. Plot each step as a visual aid. Using Mathematica, I used NMinimize to tell me how close the tangent plane was to sphere #1. When I was satisfied with the results, I took the coordinates as the tangent point. Repeat this with the other 2 sphere equations and the tangent plane equation to obtain the other 2 tangent points.

Regards,

Bill W.