  We consider regions obtained from 120 degree rotationally invariant hexagons
by removing a core and three equal satellites (all equilateral triangles) so
that the resulting region is both vertically symmetric and 120 degree
rotationally invariant, and give simple product formulas for the number of
their lozenge tilings. We describe a new method of approach for proving these
formulas, and give the full details for an illustrative special case. As a
byproduct, we are also able to generalize this special case in a different
direction, by finding a natural counterpart of a twenty year old formula due to
Ciucu, Eisenk\"olbl, Krattenthaler and Zare, which went unnoticed until now.
The general case of the original problem will be treated in a subsequent paper.
We then work out consequences for the correlation of holes, which were the
original motivation for this study.
