Knowra Clifford's circle theorems Clifford's circle theorems Clifford's circle theorems describe incidence patterns in which successive intersections of circles produce new sets of concyclic points. They include the result that the final points in a Clifford configuration lie on one circle.
Clifford configuration : A geometric arrangement of points, lines, and circles whose incidences follow Clifford's chain pattern. It displays the successive intersections that lead to the theorems' concyclicity conclusions.
Circumcircle : The unique circle passing through the three vertices of a nondegenerate triangle. Circumcircles of triangles supply the circles repeatedly intersected in these configurations.
William Kingdon Clifford : An English mathematician and philosopher whose work ranged across geometry, algebra, and physics. The circle theorems bear his name and reflect his interest in geometric configurations.
Möbius transformation : A fractional linear map of the extended complex plane that sends circles and lines to circles or lines. It preserves generalized-circle incidences and can simplify configurations before applying the theorems.
Pascal's theorem : A theorem stating that opposite sides of a hexagon inscribed in a conic meet in three collinear points. It is a celebrated incidence theorem, but its construction uses conics and collinearity rather than circle chains.
Complete quadrilateral : A figure formed by four lines, with six intersection points and four triangles. Its four circumcircles provide the starting arrangement for Miquel's theorem.
Circle intersection : The set of points shared by two circles, ordinarily containing zero, one, or two points. Successive shared points generate each stage of a Clifford configuration.
Clifford's parallel theorem : A theorem extending circle-incidence patterns to configurations involving spheres in higher-dimensional spaces. It is a related result associated with Clifford's broader study of geometric chains.
Complex plane : The plane whose points are represented by complex numbers. Complex coordinates encode circles and their intersections in compact algebraic form.
Desargues' theorem : A theorem stating that corresponding sides of perspective triangles meet on one line. Its incidence conclusion is collinearity, contrasting with Clifford's concyclicity.
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