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Axiom 4. The three diagonal points of a complete quadrangle are never
collinear.
- A complete quadrangle is a set of four points, no three of which
are collinear, and the six lines incident with each pair of these points. The
four points are called vertices and the six lines are called
sides
of the quadrangle.
Example. Complete quadrangle ABCD has vertices A, B, C, and D.
The sides are AB, AC, AD, BC, BD, and CD.
- Two sides of a complete quadrangle are opposite if the point
incident to both lines is not a vertex.
Example. Complete quadrangle ABCD has three pairs opposite sides AB
and CD, AC and BD, and AD and BC.
- A diagonal point of a complete quadrangle is a point incident with
opposite sides of the quadrangle.
Model. The vertices of the quadrangle ABCD are A, B, C and
D.
The sides of the quadrangle are AB, AC, AD, BC, BD, and CD.
The opposite sides are AB
and CD, AC and BD, and AD and BC. The diagonal
points of the quadrilateral are E, F and G.
The vertices A, B,
C, and D of the complete quadrangle may be dragged to change the quadrangle.
Do the diagonal points remain noncollinear?
Timothy Peil, 4 February 2013, Created with GeoGebra
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