The Construction: Observation Two

We can apply the notion of duality to what we realized in observation one. In observation one we found that we could apply Pappus' theorem six times to each pair of triples of collinear points in the Projective Plane. The results are six new lines in the plane and we found that of these six lines, there are two sets of three of them which are coincident.From what we know of duality, two pairs of triples of coincident lines is dual to two pairs of triples of collinear points. Thus, we can apply the Dual to Pappus' theorem to these newly constructed triples of coincident lines and obtain a point in the plane. Moreover we can permute the construction again and we will obtain six new points in the plane. We can apply the dual to observation one and it is obvious that these six points can be broken up into two pairs of three collinear point.

We will explicitly state the construction thus far. Let x=(X1 ,X2,X3)and y=(Y1,Y2,Y3) be two sets of three collinear points. Then let \(l_{1}=\rho(x,y)\) , \(l_{2}=\rho(\sigma x,y)\) , \(l_{3}=\rho(\sigma^{2} x,y)\) , \(m_{1}=\rho(\tau x,y)\) , \(m_{2}=\rho(\sigma \tau x,y)\) ,and \(m_{3}=\rho(\sigma^{2} \tau x,y)\) be the lines constructed by permutations of Pappus' Theorem. Observation one tells us that l1, l2, and l 3are coincident and the lines m1, m2 ,and m3 are coincident as well. Let l=(l1,l2,l3)and m=(m1, m2,m3)be triples of coincident lines. Then we can apply the dual to Pappus' theorem six times to l andm . Let \(X'_{1}=\rho^{ast}(l,m)\) , \(X'_{2}=\rho^{ast}(\sigma l,m)\) , \(X'_{3}=\rho^{ast}(\sigma^{2} l,m)\) , \(Y'_{1}=\rho^{ast}(\tau l,m)\) , \(Y'_{2}=\rho^{ast}(\sigma \tau l,m)\) ,and \(Y'_{3}=\rho^{ast}(\sigma^{2} \tau l,m)\) .Then by the dual to Observation One, X'1, X' 2,and X'3 are collinear, as are Y'1 , Y'2,and Y'3.

We can make another observation in addition to results provable from observation one. 

Observation 2   The triples of points return to the same line. That is the six pointsX1, X 2,X3,X'1,X'2, andX'3 all lie on the same line as do Y1 ,Y2,Y3,Y'1, Y '2, andY'3.

An applet demonstrating this observation is shown below. 

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Again, you may move the green and the blue points around the screen. The red points indicate the points X'1 , X'2,X'3,Y'1,Y '2, andY'3.

The Final Step: Observation 3