Theorem. Every orientation-preserving isometry of R^{2} with a fixed point is a rotation.
Proof. Let f be a given orientation-preserving isometry of R^{2} with fixed point c. Let tau be “translation by c”, i.e., tau(x) = x + c. Then the isometry g = tau^{-1} \circ f \circ tau has the property g(0) = 0. Since g is an affine map that fixes the origin, g must be a linear transformation of R^{2} that is distance-preserving. Therefore, g(x) = U x for some 2 \times 2 orthogonal matrix U. By an exercise in the previous assignment U must be one of the matrices formed using cos theta and sin theta for some value of theta, and since g is orientation-preserving, det U > 0 with the result that U must be the specific matrix
| . |
Therefore, g is the rotation about the origin through the angle theta, and f is the rotation about the point c through the angle theta.
Theorem. Every orientation-reversing isometry of R^{2} with a given fixed point is the reflection in some line containing the fixed point.
Proof. The argument is very similar to the preceding argument except that the 2 \times 2 orthogonal matrix U satisfies det(U) < 0 since the isometry is orientation-reversing, and, therefore,
| U = |
| , |
which is the matrix of reflection in the line through the origin forming the angle theta/2 with the positive first coordinate axis.
Proposition Every rotation of R^{2} is the composition of the reflections in two lines passing through its center.
Proof. For example, let sigma_{1} be reflection in the horizontal line through the center and let sigma_{2} be reflection in the line through the center forming angle theta/2 with the horizontal, where theta is the angle of rotation about the center. Then sigma_{2} \circ sigma_{1} is the given rotation.
Prove: If an isometry f of the plane is a rotation about the point p, then for every point x in the plane p must lie on the perpendicular bisector of the line segment from x to f(x).
Show that every translation of R^{2} is the composition of the reflections in two parallel lines that are perpendicular to the direction of translation.
Show that the composition of a rotation with the reflection in a line through the center of the rotation is another such reflection.
Let A, B, C, and P be four points in a plane, no three of which are collinear. Let PA meet BC at D, PB meet CA at E, and PC meet AB at F. Prove that D, E, and F are not collinear.