Final answer:
Euclid's proof of the Pythagorean theorem using hyperbolic geometry principles would involve non-Euclidean axioms and potentially trigonometric functions, but not the classic Euclidean theorem.
Step-by-step explanation:
Euclid's proof of the Pythagorean theorem in hyperbolic geometry would not involve the theorem as we understand it in Euclidean geometry, as hyperbolic geometry is a form of non-Euclidean geometry. In such geometries, the internal angles of triangles do not add up to 180 degrees, and the notion of parallel lines differs from that in Euclidean space. The proof in hyperbolic geometry would instead be based on the principles and axioms of that geometry, which are different from Euclid's postulates. Moreover, in hyperbolic geometry, the relationship between the sides of a triangle is not given by the classic Pythagorean theorem (a² + b² = c²), and would instead involve more complex relationships that can include trigonometric functions depending on the chosen model of hyperbolic space.