The Construction: Observation Three

Observation Three is an independence result about the two new triples of points constructed in Observation Two. 

Let x =(X1,X2,X3) and y=(Y1,Y2,Y3) be two sets of three collinear points that lie on the line l andm respectively. Let z=(Z 1,Z2,Z3) be a triple of collinear points that lie on the linen, where n is a line such that \(n \cap l = m \cap l\) that is all three of these lines are coincident. The line n need not be distinct from m.

Observation 3   The triple of points on the line l constructed from the triples of pointsx and yby the construction described in Observation Two is exactly the same as the triple of points constructed using the triples of points x and z.
 

An applet demonstrating Observation Three is shown below: 

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The green and the blue points in the applet above are movable. However, the green points are fixed to remain on the same horizontal line and the blue points must lie on a line that does not change the intersection point of the two lines. Notice that wherever the blue points are moved,the red points that lie on the same line as the green points are not affected.

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