Primitive terms: point, line

I.1: Given two different points, the points lie on one and only one line.

I.2: Each line contains at least three distinct points.

I.3: There are at least three noncollinear points.

Perpendiculars:

Primitive terms: perpendicular

P.1: Let l and m be lines. If l is perpendicular to m, then m is perpendicular to l.

P.2: If P is any point and l is any line, then there is a line m containing P such that l is perpendicular to m. If the point P is on l, then m is uniquely determined.

P.3: If l is perpendicular to m, then l and m intersect in a single point.

Motions:

Definition: Let l be a line. A reflection R_l (in l) is a one-to-one mapping of the plane onto itself with the following properties:

a. Lines are mapped onto lines.

b. If two lines are perpendicular, then their reflected images are perpendicular.

c. Every point of l is mapped onto itself.

d. There is a point of the plane which is not mapped onto itself.

For a point P in the plane, we let R_l(P ) denote the image P under R_l. The line l is called the axis of R_l.

M.1: Every line is the axis of one and only one reflection.

M.2: If R_l maps the point P to the point P ^', then R _l maps P ^' onto P .

Definition: A mapping M from the plane to itself is called a rotation if there are two intersecting lines u and v such that M = R_uR_v (i.e., M is the composition of two reflections). The point where u and v intersect is called a center of the rotation.

M.3: Consider a rotation M with center P . If l is a line containing P , then there is a reflection R_m, with P on m, that R_m(Q) = M(Q) for every Q on l.

Definition: A mapping M from the plane to itself is called a translation if there are two nonintersecting lines u and v both perpendicular to a line t such that M = R_uR_v (i.e., M is the composition of two reflections R_u and R_v) In this case M is called a translation along t.

M.4: Consider a translation M along a line t. If l is a line perpendicular to t, then there is a reflection R_m with m perpendicular to t, such that R_m(P ) = M(P ) for every P on l.

Parallels

Euclid: Given a line l and a point P not on l, there is a unique line containing P that does not intersect l.

Elliptic: Any two lines intersect.

Hyperbolic: Given a line l and a point P not on l, there are at least two different lines containing P that do not intersect l.

Betweenness and Separation

Primitive terms: between. P - Q - R denotes Q is between P and R

B.1: Let P, Q, and R be points. If P - Q - R, then R - Q - P .

B.2: Given three distinct collinear points, exactly one is between the other two.

B.3: Any four points can be named in an order P, Q, R, and S in such a way that P - Q - R - S.

B.4: If P and R are any two points, then a) there is a point X such that P - R - X and b) there is a point Y such that P - Y - R.

B.5: If P, Q, and R are points and P -Q-R, then P, Q, and R are collinear.

B.6: If R_m is a reflection and P, Q, and S are points such that P - Q - S, then R_m(P ) - R_m(Q) - R_m(S)

Plane Separation: Given a line and a plane containing it, the set of all points in the plane can be written as the disjoint union of the line and two convex sets with the property that if P belongs to one of the convex sets and Q belongs to the other, then the line segment with endpoints P and Q intersects the line.