Miquel Five Circles Theorem


Let five circles with concyclic centers be drawn such that each intersects its neighbors in two points, with one of these intersections lying itself on the circle of centers. By joining adjacent pairs of the intersection points which do not lie on the circle of center, an (irregular) pentagram is obtained each of whose five vertices lies on one of the circles with concyclic centers.

Let the circle of centers have radius r and let the five circles be centered and angular positions theta_i along this circle. The radii r_i of the circles and their angular positions phi_i along the circle of centers can then be determined by solving the ten simultaneous equations


for i=1, ..., 5, where phi_0=phi_5 and r_0=r_5.

See also

Five Disks Problem, Miquel Circles, Miquel's Pentagram Theorem, Pentagram

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Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 151-152, 1888.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. Middlesex, England: Penguin Books, p. 79, 1991.

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Miquel Five Circles Theorem

Cite this as:

Weisstein, Eric W. "Miquel Five Circles Theorem." From MathWorld--A Wolfram Web Resource.

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