Determinant Identities and the Geometry of Lines and Circles

The focus of this note is the nontrivial determinant identities which typically underlie the complex analytic proofs of all the results in the plane geometry of lines and circles. After setting up a basic dictionary relating lines and circles to complex determinants we derive such identities in connection with four geometry problems: the Steiner line, a variant of Euler’s nine-point circle, the Johnson-Tzitzeica circles, and an extension of a certain geometry problem, proposed at the 52nd International Mathematical Olympiad, Amsterdam 2011.