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.
Read more about this topic: Pappus's Hexagon Theorem
Famous quotes containing the words statements and/or theorem:
“He admired the terrible recreative power of his memory. It was only with the weakening of this generator whose fecundity diminishes with age that he could hope for his torture to be appeased. But it appeared that the power to make him suffer of one of Odette’s statements seemed exhausted, then one of these statements on which Swann’s spirit had until then not dwelled, an almost new word relayed the others and struck him with new vigor.”
—Marcel Proust (1871–1922)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)