Miquel's Point and Miquel's Thm

 

If four general lines are drawn in a plane, they will intersect at six points forming four triangles taken three at a time. In the figure the six points are A, B, C, D , E, and F. If the four triangles circumcircles are drawn, the circles will all intesect in a single pointThis point is called the Miquel Point. Only a single parabola can be drawn tangent to all four lines, and the focus of the parabola is the Miquel point.

The point is named for Augueste Miquel whose work was published in Liouville's Journal in 1838. As so often happens in math and science, however, Miquel was not the first to publish the theorem. Jakob Steiner had published it about ten years (1827-1828) earlier in Gergonne's Journal. But Steiner may not have been the first either. Antreas Hatzipolakis recently posted a note to a math history web site which suggested that William Wallace (1768-1843) may have published the theorem as early as 1799. Wallace may thus be doubly foiled, as he is known to have published a work on what is now called Simson's line prior to Simson.

Miquel's Theorem is a seperate work but also related to intersections of circles.

A circle is drawn and four points marked A, B, C, and D are placed on the circle in no particular fashion. Circles are constucted that pass through these points in pairs as shown in the figure (the third point to determine each circle is free). The four circles will also intersect in four additional points, and these points will also form a circle. The symmetry of the figure shows that if we began our construction on the four points of the interior circle, the other points of intersection would be the points A, B, C, and D.

Augueste Miquel also published an interesting theorem related to convex pentagons, and the pentagrams formed from them. An excellent prestentation of the theorem, and its proof is found at Geometry Step by Step from the Land of the Incas, a web page by Antonio Gutierrez. The site does require Flash Player, and uses lots of very current java technology.
Antonio has also added a nice interactive java applet, shown in the illustration above, which allows the theorem to be explored step by step to help make it clear. This too takes a little time to load, but is well worth the wait.

The Pentagon theorem may be viewed as an extension of the theorem of four lines at the top of the page. If the five lines (forming the pentagram) are considered in groups of four, they will produce four points where the groups of circles intersect. These four points lie on a circle themselves.

In the collected works of W. K. Clifford, edited by Robert Tucker, Clifford extends this patter to show, in his words, "the series in interminable; that is that 2n lines determine 2n circles all meeting in a point, and that for 2n+1 lines the 2n+1 points so found lie on the same (a common, I think he intends) circle.