There are 4 solid spheres arranged so that each one is touching all of the others. The 3 bottom spheres touch the flat floor at points A, B and C. The top sphere has a radius of 12 centimeters. If it were replaced by a sphere with radius 25 cm, then its center would be 14, 15 and 16 cm further from from points A, B and C, respectively.
What is the radius of each sphere?
I decided to share what I've tackled so far.
If two spheres of radius a and b are sitting on a floor touching each other, the distance from the points they touch the floor is AB=2sqrt(ab).
For 3 spheres, their floor points then form a triangle ABC with 3 known side lengths.
Then angles of this triangle can be found with the law of cosines: cos(BAC)=(ac+abbc)/(2a sqrt(bc)).
These three spheres can then be given equations. Here's a top view with centers at (0,0,a) (2sqrt(ab),0,b) (messy,messier,c):
https://www.desmos.com/calculator/7byt50y5me
I don't know how to get the 4th sphere to sit on top.
Once I figure this out it will be a lot more algebra.

Posted by Jer
on 20190207 21:30:14 