What exactly is a quadrilateral?

"Four points joined in order: is that always a quadrilateral?" Not as we picture one. The book's odd shapes show what can go wrong:

• three corners on one line: a triangle (NOPE) or sides lying on top of each other (SILY);
• one corner lifted out of the flat plane: a non-planar 4-gon (BENT), like two pens held so they neither meet nor run parallel;
• two sides crossing each other: a self-intersecting 4-gon (CUTS).

The careful definition. Four different points A, B, C, D in a plane, no three on one line, such that every point of AB, BC, CD and DA (other than the corners) lies on exactly one of these sides.

The words:

• adjacent sides are neighbours: they share a corner (AB and BC);
• opposite sides have nothing in common (AB and CD);
• AC and BD are the diagonals;
• ABCD has 8 names: start at any corner and go either way round.

Convex or dented? A quadrilateral is convex when every inside angle is less than 180°, or, the same thing, when its diagonals cross. DART, with a dent at D, is non-convex. A quick test: if one corner lies inside the triangle made by the other three, there is a dent there. For example, A(0, 0), B(2, 0), C(1, 1), D(1, 3) has C inside triangle ABD, so ABCD is dented at C.

Where can the fourth corner go? Fix A, B and C. The lines AB, BC and CA cut the plane into 7 regions: D gives a convex 4-gon in 1 of them, a crossed one in 2, and a dented one in 4.

Crossed 4-gons are useful. They appear in real machines: four-bar linkages pass through self-intersecting shapes as they move.