5-Star and A Circle: What Is It About?
A Mathematical Droodle

Explanation

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Copyright © 1996-2017 Alexander Bogomolny

A "common" 5-star defines five triangles and a pentagon. (It could be built starting with a pentagon whose sides are the extended to intersect at the vertices of a star.) Each of the five triangles is inscribed in a circle. The neighboring circles intersect at two points: a vertex of the pentagon and one other point. It's remarkable that those "other" points lie on a circle.

Clifford circle on a star

The above statement falls into a very general framework of center circle chains. Four lines define four triangles whose circumcircles meet at a point, known as Miquel's point of the 4-line. Five lines define five Miquel's points that are known to lie on a circle, Clifford's Circle of the 5-line.

The 5-star is defined by 5 lines. When one line is disregarded, the remaining four cross in such a manner as to form (among other polygons) two adjacent triangle legs of the star. Obviously, the circumcirlces of the latter intersect at Miquel's point of the 4-line. This proves the statement.

However, there are also simple direct proofs. To be continued ...

References

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Copyright © 1996-2017 Alexander Bogomolny

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