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This document provides a comprehensive study of **Euclidean and Affine Geometry**, covering the fundamental concepts, definitions, properties, and applications of geometric structures. It explores Euclidean geometry, including points, lines, planes, angles, triangles, circles, distance, congruence, similarity, and coordinate geometry. It also introduces Affine Geometry, focusing on affine spaces, affine transformations, parallelism, vectors, translations, and geometric invariants. The document includes theoretical explanations, mathematical proofs, worked examples, diagrams, and practice problems to strengthen understanding. It is designed as a university-level resource for students studying Mathematics, Graph Theory, Geometry, Computer Science, or related engineering courses, making it suitable for exam preparation, assignments, and academic reference.
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Take three different examples that are homomorphic but not isomorphic give reasons as well
Explain the isometries of the plane using different AI tools such as ChatGPT, Deep Seek, and Gemini or identify the differences in the data produced by these AI tools.