### Six Points, Three lines

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

Given three lines *a*_{0}, *a*_{1}, *a*_{2} that intersect in three points _{0} = *a*_{1} ∩ *a*_{2},_{1} = *a*_{0} ∩ *a*_{2},_{2} = *a*_{0} ∩ *a*_{1},_{0} on *a*_{0}, A_{1} on *a*_{1}, A_{2} on *a*_{2}.

Then the circumcircles of triangles A_{0}A_{1}D_{2}, D_{0}A_{1}A_{2}, and A_{0}D_{1}A_{2} are concurrent (Pivot theorem). If, in addition, points A_{0}, A_{1}, and A_{2} are collinear, then circle D_{0}D_{1}D_{2} 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.

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 A_{i} can be dragged along the corresponding lines.

|Activities| |Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 Alexander Bogomolny