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# 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 and let the five circles be centered and angular positions along this circle. The radii of the circles and their angular positions along the circle of centers can then be determined by solving the ten simultaneous equations

 (1) (2)

for , ..., 5, where and .

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

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## References

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.

## Referenced on Wolfram|Alpha

Miquel Five Circles Theorem

## Cite this as:

Weisstein, Eric W. "Miquel Five Circles Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MiquelFiveCirclesTheorem.html