Pappus's Hexagon Theorem - Other Statements of Pappus's Theorem

Other Statements of Pappus's Theorem

In addition to the above characterizations of Pappus's Theorem and its dual, the following are equivalent statements:

• If the six vertices of a hexagon lie alternately on two lines, then the three points of intersection of pairs of opposite sides are collinear.

• In a matrix of 9 points (as in the picture and description above), if the first two rows and the six "diagonal" triads are collinear, then the third row is collinear. That is, if ABC, abc, AbZ, BcX, CaY, XbC, YcA, and ZaB are all lines, then Pappus's theorem states that XYZ must be a line. Also, note that the same matrix formulation applys when A, B, C and a, b, c are concurrent lines in the dual form of the theorem.
• Given three distinct points on each of two distinct lines, pair each point on one of the lines with one from the other line, then the joins of points not paired will meet in (opposite) pairs at points along a line.
• If two triangles are doubly perspective, then they are trebly perspective.
• If AB, CD, and EF are concurrent and DE, FA, and BC are concurrent, then AD, BE, and CF are concurrent.