  The enumeration of lozenge tilings of hexagons with holes has received much
attention during the last three decades. One notable feature is that a lot of
the recent development involved Kuo's graphical condensation. Motivated by
Ciucu, Lai and Rohatgi's work on tilings of hexagons with a removed triad of
bowties, in this paper, we show that the ratio of numbers of lozenge tilings of
two more general regions is expressed as a simple product formula. Our proof
does not involve the graphical condensation method. The proof is short and
direct. We also provide a corresponding formula for cyclically symmetric
lozenge tilings. Several previous results can be easily deduced from our
generalization.
