I am a student, too. Let's go to recess, ok? These questions and tasks are for the Math People.
I say, Mr Borcherds' worksheet http://www.geogebratube.org/material/show/id/3597, alas, is not a shape of constant width.
Proof: drag the given points.
The only purpose I have for finding errors is to fix them. So let me try my hand at a solution:

I say, the black curve through P and Q is a curve of constant width (diameter).
I included a Hint above for how, following Euler, we might define curves of constant width in general. For this worksheet, I followed Mr. Borcherds construction, applying constraints and corrections as they arose. Let me step through his worksheet http://www.geogebratube.org/material/show/id/3597, and my thought process.
I divide "not a curve of constant width" into two CASES:

Manipulation of the given points changes the diameter: Let one point pass another. Then
a) the figure changes size. The question remains, how to construct a family of figures of constant diameter?Q: Does such a family exist? If so, what are the constraints?
b) the new figure no longer has a distinct diameter in each direction. By definition, this is not a curve of constant width.
Assertion: There exists a curve of constant width from the same generating points, with the new (maximum) diameter.
(Task: Construct such a figure.)

The points can be manipulated so that the figure has a variable (maximum) diameter.
(Proof: Drag the given points so that the dashed figure is convex.)
Assertion: I say, there exists such a figure of constant diameter, from the same given points and construction method.
Task: (Construct such a figure.)

I found the following questions and puzzles pertinent:
What are the constraints on the arcs/moving points?
In GGB, I have given every arc by one of two custom methods. Why?
Suppose I wish to rotate the figure in place, in a fixed box about a fixed pivot. I will cut a groove into the figure, and let it slide along the pivot. What is the shape (locus) of the groove?
I have passed a diameter through the figure, rather than using a bounding box. Why?
Hints and Challenges:

The figure is handed: Given vector AB, point C gives the direction of construction. Suppose I begin with C on the other side of AB. What is the maximum magnitude of arc AC, on the circle with center B?

Click the black curve, then drag (for example) C until it coincides with E.
What is the magnitude of ∡CDE? ... Arc CE?
..and ∡EDC? ...and arcEC?
Demonstrate your answer, using the given figure (above).
Now demonstrate the answer in your own construction.
Does GGB agree?

What happens to point C when you try to pull it past A?

Let a circle pass through three given points K, L, M.
What is the length of the arc in the limiting figure as K, L, M approach one another, and coincide?
Demonstrate your answer with GGB.

In the figure above, consider the arc from C to A, on the circle with center B and radius r. When is | arcCA| > π r?
Demonstrate your answer with your own construction in GGB.

Extra Credit:In angular measure, what is the mathematical relationship between orientation (heading) and arc length?
How can this relationship be made perfectly definite?
Though proofs were omitted, I have provided working demonstrations of correct answers to all the questions and tasks.
_________
I disapprove of trick questions. Let me be plain: the angle system in GGB is pretend.
I know how to measure anyway. Lucky for me, I guess.
Consider always the student.