A novel honeycomb liquid-crystalline (LC) phase with triangular cells is found for a new compound composed of three incompatible molecular parts, namely, a rigid terphenyl core, two different nonpolar alkyl chains in the terminal 4- and 4' '-positions, and a polar flexible oligo(oxyethylene)chain terminated by a COOH group in the lateral 3'-position. The self-organization of this compound was studied by polarized microscopy, differential scanning calorimetry, and X-ray diffraction. The triangular LC phase is based on a hexagonal 2D-lattice, plane group p6mm. The phase is built up of triangular cylinders, each framing a column of polar side chains. The alkyl chains form separate columns, each connecting six aromatic cylinder walls. The present triangular phase is the topological dual of the hexagonal honeycomb cylinder structure reported earlier, and in the series of polygonal honeycomb structures its polygons have the minimum number of sides.