Six Points, Three lines

The applet supplies an extra illustration of Miquel's theorem as well as Pivot Theorem.

Given three lines a0, a1, a2 that intersect in three points D0 = a1a2, D1 = a0a2, D2 = a0a1, and points A0 on a0, A1 on a1, A2 on a2.

Then the circumcircles of triangles A0A1D2, D0A1A2, and A0D1A2 are concurrent (Pivot theorem). If, in addition, points A0, A1, and A2 are collinear, then circle D0D1D2 is concurrent with the other three (Miquel's theorem). The point of concurrency is the Miquel point of the quadrangle (complete quadrilateral, 4-line) formed by the four lines.

 

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

In the applet, the lines can be dragged by their end points (to rotate about the other end) or anywhere in-between to translate parallel to their position. Points Ai can be dragged along the corresponding lines.

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