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# Constant width for MBorcherds

- Author:
- Ryan Hirst

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.)

- 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.