Analytic Foundations of Geometry

Robert S. Wilson

4. Translations and Reflections

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.

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Theorem 4.6: The reflection   R   of the point   A   in a line is

R(A) = 2F - A

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

R(x1, y1) = (2a - x1, y1)

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.

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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

T(x, y) = (x - x0, y - y0)

then the translation

U(x, y) = (x + x0, y + y0)

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.

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5. Arc Lengths, Angles, and Rotations