Section 9.3 Quadrilaterals
¶What is a quadrilateral? Take a moment to think about your answer before reading further.
A triangle consists of three sides and three angles. But what is a side? There are no line segments in projective geometry, nor are there angles.
Let's try again. A triangle consists of three (non-concurrent) lines, together with their intersection points. That definition seems OK, but a better term for this geometric object would be trilateral. We can continue to refer to the “sides” of projective polygons, so long as we remember that they are lines, rather than line segments.
So, what's a triangle? In Euclidean geometry, there is an angle at each vertex, so it's not much of a stretch to adopt the language triangle to refer to three (non-collinear) points, together with the lines joining them.
It's easy to see that, with these definitions, triangles and trilaterals are the same, in both cases consisting of three points and three lines.
So what is a quadrilateral? Well, we can take four lines, together with their intersection points. But which intersection points? Figure 9.3.1 shows two different quadrilaterals, with the same lines; order matters. We adopt the notation \(\square abcd\) for the quadrilateral containing the intersection points of \(a\) with \(b\text{,}\) \(b\) with \(c\text{,}\) \(c\) with \(d\text{,}\) and \(d\) with \(a\text{,}\) as shown in the first example in Figure 9.3.1; the second example is then \(\square acbd\text{.}\)
We can also define a quadrangle as four points, together with the lines connecting them. Again, order matters. Figure 9.3.2 shows two different quadrangles, with the same vertices. In analogy with the notation for quadrilaterals, we adopt the notation \(\square ABCD\) for the quadrilateral containing the lines connecting \(A\) with \(B\text{,}\) \(B\) with \(C\text{,}\) \(C\) with \(D\text{,}\) and \(D\) with \(A\text{,}\) as shown in the first example in Figure 9.3.2; the second example is then \(\square ACBD\text{.}\)
We can avoid worrying about order if we instead consider all of the intersections. So a complete quadrilateral consists of four lines and all six points of intersection, and a complete quadrangle consists of four points and all six lines connecting them, as shown in Figure 9.3.3.