# k-Equivalence Relations and Associated Algorithms

Lines and circles pose significant scalability challenges in synthetic geometry. A line with n points implies n 3 collinearity atoms, or alternatively, when lines are represented as functions, equality among n 2 different lines. Similarly, a circle with n points implies n 4 cocyclicity atoms or equality among n 3 circumcircles. We introduce a new mathematical concept of k-equivalence relations, which generalizes equality (k=1) and includes both lines (k=2) and circles (k=3), and present an efficient proof-producing procedure to compute the closure of a k-equivalence relation.

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