The Quest for the Lost Quadrilateral

Recently, David Griswold posted this in #geomchat:

I was immediately struck by the diagram's asymmetry – even though it contains all standard types of convex quadrilaterals. As if a line connecting trapezoid and kites were missing, which of course cannot be. So what else is?

The apparent skewness of the classification is especially insulting given the beautiful symmetry in its lower part, where rectangles and rhombuses exhibit a deep duality between lengths and angles. Let's revel in it for a brief moment before we tackle the larger picture.

Classifying parallelograms

A parallelogram, as you might recall, is this kind of quadrilateral:

All four properties are logically equivalent, meaning that either of them can be taken as the defining property, and the others derived from it. (Personally, I prefer the first one – nomen est omen.)

Rectangles are a special class of parallelograms:

The angles at the vertices are not only pairwise equal, but as all four as well (and thus all are 90°). It follows that the diagonals, in addition to still meeting in their common midpoint, have equal length. This last property is often not explicitly mentioned in geometry class, but implicitly used in application problems. Again, one could go the other way around and use the diagonal property as the definition and derive (how?) the right-angle property.

Another special class of parallelograms are the rhombuses:

In a rhombus, all four sides have the same length, not only as opposite pairs. A consequence of this is that the diagonals meet at a right angle (think congruence). There is also the noteworthy property that the diagonals bisect the angles, which we will get back to shortly.  Finally, the classes of rectangles and rhombuses intersect to form the class of squares.

Have you noticed something peculiar? Because I have.

As @JamesTanton likes to put it: "Whoa."

Take it in slowly. Revel in it. This is, hands down, one of my favourite chapters in geometry.

Compare the rectangle and rhombus for a while longer. Notice how any symmetry of lengths in one mirrors a corresponding symmetry of angles in the other?

Now, why would such a piece of exquisite beauty sit right next to a helter-skelter classification of the other quadrilaterals? This would be unbearingly insulting to the mathematician's mind. Hence I sought out to find the missing piece, letting the duality between lengths and angles be my guide.

Extending the duality

Look at the kite. It shares some commonality with the rhombus, in that the diagonals meet at right angles, but only one bisects the other. In addition, one of the diagonals acts as a bisector of a pair of opposite angles, whereas in the rhombus both diagonals do.

Now look at the trapezoid and its special case, the isosceles trapezoid. Regarding their diagonals and angular bisectors, not much of interest is going on. But bear in mind the duality between angles and lengths, or rather sides. What is there of notice about a trapezoid's sides?

• In a trapezoid, a pair of opposite sides is parallel. Put differently, their segment bisectors are parallel:

• In an isosceles trapezoid, these opposite segment bisectors coincide:

Let's rephrase the property of the isosceles trapezoid and compare it with the kite:

• In an isosceles trapezoid, a pair of opposite sides shares a common bisector. (It follows that they are parallel.)
• In a kite, a pair of opposite angles shares a common bisector.

The isosceles trapezoid generalizes to the trapezoid by demanding that the side bisectors be only parallel (and hence the sides themselves are parallel). This suggests a corresponding generalization of the kite, which I christen a "kitoid":

Definition. A kitoid is a convex quadrilateral in which a pair of opposite angles has parallel angular bisectors.

(The outer point serves as a "handle" to manipulate the kitoid's form.)

Then we can arrange kitoids, kites, trapezoids and isosceles trapezoids in a nifty little 2x2 diagram:

I move we call the clunkily named "isosceles trapezoid" simply "trapeze" for symmetry. (My fellow German speakers might disagree, because "Trapez" already stands for "trapezoid", but I say we take a hit for the team.)

You can easily check the following statements which embed kitoids into the greater hierarchy:

Theorem.
(a) The kitoids that are trapezoids are precisely the parallelograms.
(b) The kitoids that are trapezes are precisely the rectangles.
(c) The kites that are trapezoids are precisely the rhombuses.
(d) The kites that are trapezes are precisely the squares.

The classification of convex quadrilaterals is now breathtakingly symmetric:

(Red arrows indicate restricting definitions, black arrows intersections.)

There is a useful expression for the area of any trapezoid, trapeze or not:

However, the area formula for kites:

appears to resist easy generalization to kitoids, because of a lack of right angles. Unlike a kite, a kitoid cannot be easily doubled into a rectangle. Two possible generalizations of the kite doubling are a doubling into a parallelogram:

and one into a "rectangular ring":

(The inner rectangle has to be subtracted because of overcounting, as a consequence, the area difference between the rectangles is twice the kitoid's area.)

On the other hand, the standard constructions that accompaniy the area formula for trapezoids, such as the one above, have no obvious analogue for kitoids. It might be too much to ask for a simpler generalization of a kite's area to a kitoid. Unless the property that corresponds to the trapezoid area, under the duality between angles and sidelengths, is something different altogether.