<![CDATA[Analytic geometry functions as a cognitive bridge between visual and algebraic representations, particularly in the topics of circles and tangent lines. This qualitative descriptive study aims to map high school students’ thinking processes when solving non-routine analytic geometry problems and to identify how critical thinking and geometric imagination emerge through the coordination of visual and symbolic representations. The participants were 25 Grade XI Natural Science students at SMA Santa Maria Cirebon, selected purposively because they had studied the prerequisite material and the target topic and represented varied levels of ability. Data were collected through four non-routine essay problems, students’ written work, classroom observation, and semi-structured interviews with six selected subjects. Data analysis followed the interactive model of Miles, Huberman, and Saldaña, supported by theory-based coding indicators adapted from Facione and Ennis for critical thinking and from Duval for visual representation. The results show that only 8 of 25 students (32%) solved all problems correctly and completely. Successful students demonstrated a structured thinking flow consisting of initial visualization, algebraic reasoning, and evaluation through visual verification. In contrast, less successful students tended to use formulas procedurally without connecting geometric meaning to algebraic forms. The findings indicate that, in conventional learning, critical thinking and geometric imagination are more likely to appear when students are explicitly accustomed to constructing sketches, translating geometric conditions into equations, and checking results visually. Practically, teachers need to scaffold initial sketching and visual verification; however, because this study is limited to one class and uses a qualitative descriptive design, the findings should be interpreted as a contextual mapping of thinking processes rather than a broad claim of instructional effectiveness. Keywords: Analytic Geometry; Critical Thinking; Geometric Imagination; Circles and Tangent Lines; Visualization..]]>
