Construct the astroid.

- What is
the envelope formed by the rod?

Find the point of tangency.

Find the locus of a point carried by the rod. (A ladder standing on a smooth floor against a wall slides down. Along what curve does a cat sitting on the ladder move?)

A wooden triangle moves on a plane so that two vertices move along two fixed perpenducluar lines. How does the third vertex move?

References:

Martin Gardner: Wheels, Life and Other Mathematical Amusements,
pp.1-9.

E.H. Lockwood, A Book of Curves, Cambridge University
Press, 1962.

N.B. Vasilyer and V.L. Gutebmacher, Straight Lines and
Curves, Mir, 1980.

Robert C. Yates, A Handbook on Curves and their Properties,
Edwards, 1952.

This is related to the **Residue
Curves**, see: Graphs of Linear Congruences, Math. Scripta (147) pp.
106-112. Also see pp. 114-115, p. 224, pp. 232-233 of the same issue.