In this paper we consider arbitrary hexagons on the triangular lattice with three arbitrary bowtie-shaped holes, whose centers form an equilateral triangle. The number of lozenge tilings of such general regions is not expected --- and indeed is not --- given by a simple product formula. However, when considering a certain natural normalized counterpart of any such region, we prove that the ratio between the number of tilings of the original and the number of tilings of the normalized region is given by a simple, conceptual product formula. Several seemingly unrelated previous results from the literature --- including Lai's formula for hexagons with three dents and Ciucu and Krattenthaler's formula for hexagons with a removed shamrock --- follow as immediate consequences of our result.