[alogo] Pappus theorem

Consider three points on line a and three on b and join them as shown. Then the intersection-points A*, B*, C* of the joining lines are contained in a line c.
By applying a projectivity that sends P, Q to two points at infinity, AA'C'C becomes a parallelogram, B* becomes its center and the theorem reduces to the exercise handled in MenelausApp.html .
Considering the pair of lines (a,b) as a degenerate conic, the theorem represents a special case of Pascal's theorem.

[0_0] [0_1] [0_2]

Notice that lines a, b, c have a common point (O) exactly in the case AA', BB' and CC' are restricted to pass trhough a common point (P). In that case line c is the polar of P with respect to the two lines a and b. The following figure illustrates this.
Hint: triangles AA'C*, CC'A* are then point-perspective and the theorem of Desargues applies etc..(see Desargues.html ).

[0_0] [0_1] [0_2]
[1_0] [1_1] [1_2]

Pappus line c is simply the line-homography-axis of a homography relation, defined by A, B, C on line a and their images A', B', C' on line b. This point of view is discussed in LineHomographyAxis.html .

See Also


Return to Gallery

Produced with EucliDraw©