Mathematical pictures go beyond usual artistic representations because they often contain a great deal of information. They are sophisticated creatures that contain symbols as well as lines, angles, projections, measures,etc. In this chapter we illustrate the technique of introducing special “marks” in the pictures toidentify relevant parts: equality of segments, equality of angles, repetitions, similar or congruent subsets, etc. In many cases appropriate identification of key elements readily yields a proof of the desired result. This is also the case in Euclidean geometry where, using straightedge and compass, one must construct figures using a collection of related elements (sides, angles, bisectors,…). The procedure becomes a process of identifying how the data determine the unknown parts. In making mathematical drawings … details matter!

On the angle bisectors of a convex quadrilateral

In a triangle the three angle bisectors meet at the incenter. What happens in a convex quadrilateral? The following result gives the complete answer, and the proof is based on a simple picture in which all the relevant angles are identified.

We can make a simple picture including the basic elements described in the above statement, and we mark on it the key angles (see Figure 5.1).