4.3
Investigate the Dual of
Axiom 4
and the Definition of a Complete Quadrilateral
I hope that posterity will judge me
kindly, not only as to the things which I have explained, but also to those
which I have intentionally omitted so as to leave to others the pleasure of
discovery.
—
Rene Descartes (1596–1650)
Dual of Axiom 4. The three diagonal lines of a complete quadrilateral are never
concurrent.
- A complete quadrilateral is a set of four
lines, no three of which
are concurrent, and the six points incident with each pair of these lines. The
four lines are called sides and the six
points are called vertices
of the quadrilateral.
Example. Complete quadrilateral abcd has sides a, b, c, and d.
The vertices are a · b, a
· c, a · d, b
· c, b · d, and c
· d.
- Two vertices of a complete quadrilateral are opposite if the
line
incident to both points is not a side.
Example. Complete quadrilateral abcd has three pairs opposite vertices
a · b
and c · d, a
· c and b · d, and a
· d and b · c.
- A diagonal line of a complete quadrilateral is a
line incident with
opposite vertices of the quadrilateral.
Model. The sides of the quadrilateral abcd are
a, b, c and d.
The vertices of the quadrilateral are E = a
· b, F =
b · c, G = c
· d, H = a · d, I = a
· c and J = b
· d.
The opposite vertices are E & G, F & H, and I &
J. The diagonal lines of the quadrilateral are EG, FH, and
IJ.
Quadrilateral
Drag the points defining the four sides of quadrilateral abcd.
Are the diagonals ever concurrent?
Timothy Peil, 5 February 2013, Created with GeoGebra
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4.3 Duality in
Projective Geometry
© Copyright 2013 -
Timothy Peil