This problem was posted at Myōjōrinji temple in Gifu Prefecture in 1865. Here is its entry at the Sangaku Archive [coming soon].

In the diagram below, show that the pin circles, and also the orange circles, have one-sixth the radius of the big circle.

One of the useful triangles in this problem is oblique (non-right). The [url=https://www.geogebra.org/m/tn4uebgj]Law of Cosines[/url] may be helpful here.