Section 1
EUCLIDEAN ISOMETRIES
Prove that every function $f$ on a set $S$ that is one-to-one and onto has a unique inverse.
Prove that the inverse of an isometry is again an isometry. (This implies that the set of isometries is closed under the inverse operation.)
Let $f, g$ be two invertible functions from a set $S$ to itself. Let $h=f \circ g$; that is, $h$ is the composition of $f$ and $g$. Show that $h^{-1}=g^{-1} \circ f^{-1}$.
Let $f, g$ be two isometries. Show that the composition $f \circ g$ is again an isometry. (This says the set of isometries is closed under composition.)
Show that isometries map circles of radius $r$ to circles of radius $r$. That is, isometries preserve circles.
Prove that the image of a triangle under an isometry is a new triangle congruent to the original.
Given an equilateral triangle $A B C$, show that there are exactly six isometries that map the triangle back to itself.
Prove Corollary 5.4.
Consider points in the plane as ordered pairs $(x, y)$ and consider the function $f$ on the plane defined by $f(x, y)=(k x+a, k y+b)$, where $k, a, b$ are real constants, and $k \neq 0$. Is $f$ a transformation? Is $f$ an isometry?
Define a similarity to be a transformation on the plane that preserves the betweenness property of points and preserves angle measure. Prove that under a similarity, a triangle is mapped to a similar triangle.
Use the previous exercise to show that if $f$ is a similarity, then there is a positive constant $k$ such that$$f(A) f(B)=k A B$$for all segments $\overline{A B}$.
Consider points in the plane as ordered pairs $(x, y)$ and consider the function $f$ on the plane defined by $f(x, y)=(k x, k y)$, where $k$ is a non-zero constant. Show that $f$ is a similarity.