Theorems 4.1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
Theorem 4.1: (The distance axioms) Let A, B, and C be any three points in the plane. Then
1. |AB| > 0 with equality if and only if A = B.
2. |AB| = |BA|
3. |AB| < |AC| + |CB| with equality if and only if C is on the line segment between A and B.
Theorem 4.2: If M is an isometry and C is a point on the line segment between A and B, then M(C) is a point on the line segment between M(A) and M(B).
Theorem 4.3: Let M be an invertible isometry1. If A and B are two points which are on the same side of a line L, then M(A) and M(B) are on the same side of M(L).2
Theorem 4.4: If we compose two isometries, the result is an isometry.
Theorem 4.5: A translation is an isometry.
Theorem 4.6: The reflection R of the point A in a line is
where F is the foot of the point A in the line.
Theorem 4.7: The reflection R of the point (x1, y1) in the line y = mx + b is
The reflection R of the point (x1, y1) in the vertical line x = a is
Theorem 4.8: A reflection is an isometry.
Theorem 4.9: If R is a reflection, then R(R(A)) = A for all points A in the plane.
Theorem 4.10: If a point on a circle is reflected about a line that goes through the center of the circle, then the image of the point is also on the circle.
Theorem 4.11: If T is the translation
then the translation
is the inverse of T.3
Theorem 4.12: If two circles intersect, then the two points of intersection are reflections of each other about the line joining their centers.
5. Arc Lengths, Angles, and Rotations