In the papers linked to below, complete solutions to the problems mentioned in the title are provided or, as in the case of two well known problems, references are provided. These problems serve as fruitful applications for classes in calculus including differential equations. No higher mathematics is required. The solutions are divided up over several papers due to the overall length. The solutions error on the side of more detail rather than less.
The main focus of these problems is the study of curves related to given curves. The short section 14.1 in Eves' book gives a very nice overview but as a good teacher he leaves the details to his readers.
There is a glossary of terms provided in the paper Part V. This glossary contains the definitions of the curves studied in these exercises. There are also appendices in some parts to provide background on selected topics that might need a little review.
The following links will cover the exercises as indicated.
1. Exercises 2,4,5,6,7 Part I - Solutions 2,4,5,6,7
2. Exercise 3 on two constructions of the center of curvature
3. Exercises 8,16,24,26,27 on curves related to equiangular spirals
4. Exercises 9,10,11,12,13,14,15 Part II - Solutions 9,10,11,12,13,14,15
5. Exercises 17,18,19,20,21,22,23,25 Part III - Solutions 17,18,19,20,21,22,23,25
6. Exercises 28,29,30,31,32,33,34,35,36 Part IV - Solutions 28,29,30,31,32,33,34,35,36
7. Exercises 37,38,39,40 Part V - Solutions 37,38,39,40
8. More details on the cycloid and its evolute
