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 = a₁ ∩ a₂, D₁ = a₀ ∩ a₂, D₂ = a₀ ∩ a₁, 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.

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| |Store|