Triangular fuzzy multiplicative preference relation (TFMPR) is a widely used preference representation framework in the fuzzy analytic hierarchy process (FAHP). In this article, we develop a triangular fuzzy multiplication-based equation to characterize transitivity among original fuzzy assessments in a consistent TFMPR. A geometric consistency index is then proposed to measure the inconsistency of TFMPRs. By capturing row fuzziness proportionalities and the difference between the row increasing and decreasing part fuzziness indices, this article establishes two logarithmic least square models to find normalized fuzzy weights from two kinds of TFMPRs. The two logarithmic least square models are further integrated into one whose analytic solution is found by the method of Lagrange multipliers. A novel method is put forward to check the acceptability of TFMPRs by both examining acceptable consistency and acceptable fuzziness. This article devises a parametric defuzzification approach to obtain the real-valued weights of criteria for aggregating the local normalized fuzzy weights into the global fuzzy weights in the triangular FAHP. A comparative analysis of the proposed model with existing fuzzy eigenvector methods and fuzzy average methods is carried out by a numerical example with four TFMPRs to clarify its validity and merits. An outstanding teacher award selection problem is provided to show the practicality of the proposed triangular FAHP.