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.