Because SketchUp's circles are in reality finely faceted polygons the real-intersection of two arcs' circumferences is unlikely to be exactly on those arcs' edges, unless by coincidence it turns out to be at one of the arc's edge's vertices.
Conversely the actual-intersection of those arcs' edges is unlikely to fall exactly on the arcs' circumferences, unless by coincidence it's at a vertex of those arcs' edges...
My TrueTangent tools contain a number of algorithms to find circles/arcs' true tangents, true intersection points etc...
Perhaps these will help you find what you need... 😕