There are good reasons why the theorems should all be easy and the definitions hard.

-Michael Spivak

In this article, we look at a selection of tasks related to the angle bisectors of a quadrilateral and discuss their potential function in GeT courses. An instructor may choose to begin an exploration of the properties of the angle bisectors by looking at the quadrilateral formed by the lines in question. But this already provides an opportunity for students to engage in thinking much like a math researcher, as well as to discover the properties of these lines. Consider the following task:

Consider the four angle bisectors of a quadrilateral Q₁. How many times can at least two bisectors intersect? What rules can consistently choose four points to define a second quadrilateral Q₂? When Q₂ exists, what is its area and when is it 0? Deduce other properties of Q₂?

At the heart of this task is the request for the student to formulate a definition. The angle bisectors of a quadrilateral could form as many as 6 distinct intersections, and it is not trivial to determine which ones are “natural” choices to determine a second quadrilateral. While the “correct” definition is to think of angle bisectors as rays oriented inward instead of lines, the open-endedness of the question nevertheless allows students to perhaps build their own justification (sound or not). An instructor could continue the class by asking students to compare the definitions they’ve created, so they can see the creative and interpretive nature of mathematics.

The next task is designed to give students a greater sense that definitions are not chosen arbitrarily, but because of their power in proving theorems. Once the class agrees to define the “induced quadrilateral by the angle bisectors” as those with vertices of the intersection of the rays, consider the following task:

Group task: Write definitions of a square, rectangle, rhombus, and kite to make the following theorems true when the area of the induced quadrilateral is 0.

• For a quadrilateral Q, Q is a parallelogram if and only if the angle bisectors of Q form a rectangle.

• For a quadrilateral R, R is a rectangle if and only if the angle bisectors of R form a square.
Traditional treatment of these theorems require that Q is not a rhombus, nor R a square. So this is an especially natural task to pose after an instructor has presented the theorems in the traditional manner. Not only does it allow the students to make the statements more elegant, but it also shows them that a theorem is occasionally improved not through an improved proof, but through a generalization of the definitions. So with these tasks, a GeT instructor can guide their students not only through Euclidean geometry but also through researchlevel mathematical thinking. A nontrivial amount of time spent in research is useful not necessarily in obtaining results about a mathematical object, but in pinpointing the abstract properties of the object that necessitate the results in question. Through these tasks, a GeT instructor can create this environment of creating definitions: an undertaking that, as Spivak indicates, can be quite difficult.

Matt Park is a research assistant in the GRIP Lab.