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.
The points
a,
b, c,
and d
may
be dragged to change the quadrilateral.
Do the diagonal lines remain nonconcurrent?
To reset to the original settings, type the letter
"R" on the keyboard.
