Problem solving is one of the high-level thinking skills and essential in teaching mathematical concepts and procedures. The eye-tracking method allows educators to see and interpret different problem-solving behaviours of students. The purpose of this study was to investigate the problem-solving behaviours of university students based on eye tracking statistics when faced with a geometry problem, and to explain these behaviours in terms of the students' success and learning styles. The research design was a case study approach that combined quantitative and qualitative data. Eight first-year students from the department of elementary mathematics education in a private University, Turkey, participated in this study. They were asked to solve a specific geometry question using the GeoGebra software. While solving this question, their eye movements were recorded and analysed using an eye-tracking program. Results were analysed to investigate their problem-solving skills and behaviours based on Polya's mathematical problem-solving stages. Results of the study suggest that the effort devoted to the understanding and planning stages of problem-solving are important factors that contribute to success in this case. Results also revealed that participated students with a converger learning style were more successful in each step of the problem-solving process. Additionally, it could be concluded that the input area had only been used by converger students, indicating that they devoted time and effort to creating and trying formulas.