Recently, automatic question generation (AQG) has been researched extensively for educational purposes. Existing approaches generally lack relevant information on the authentic context and problem diversity with various difficulty levels, so we proposed a new AQG system for generating contextualized and personalized mathematic word problems (MWP) in authentic contexts using the Generative Pre-trained Transformers (GPT). Our proposed system comprises (1) authentic contextual information acquisition through image recognition by TensorFlow and augmented reality (AR) measurement by AR Core, (2) a personalized mechanism based on instructional prompts to generate three different difficulty levels for learners' different needs, and (3) MWP generation through GPT with authentic contextual information and personalized needs. We conducted a quasi-experiment with the participation of 52 students to evaluate the effectiveness of the proposed system on geometry learning performance. Further, the learning behaviors were analyzed in the aspects of authentic context, mathematics, and reflective behavior. The findings showed better results in geometry learning performances from students who learned with contextualized and personalized MWPs than those who were taught without contextualization and personalization on MWPs. Moreover, it was found that student's ability to comprehend the practical situation or scenario presented in a problem (problem context understanding) and students' ability to recognize relevant information from the problem context (identifying contextual information) significantly improved their learning performance. Moreover, students' ability to apply math concepts and solve medium-level MWP also contributes to the improvement of learning performance. Meanwhile, learners showed positive perceptions toward the proposed system in facilitating geometry learning. Therefore, it is useful to promote an authentic context setting for mathematical problem-solving.