Theorem 2.1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14
Theorem 2.1: (The parametric representation of a line) Given two points (x1, y1) and (x2, y2), the point (x, y) is on the line determined by (x1, y1) and (x2, y2) if and only if there is a real number t such that
and
Theorem 2.2: (The parametric form of the Ruler Axiom) Let t be a real number. Let A and B be two points. Let
and
Let
using vector addition and scalar multiplication of points. Then
and
Theorem 2.3: If C is on the line segment between A and B then
If C is on the line determined by A and B but on the other side of B from A then
If C is on the line determined by A and B but on the other side of A from B, then
Corollary: (The midpoint formula) Let (x1, y1) and (x2, y2) be two points. The midpoint between them has coordinates
Theorem 2.4: If C is on the line segment between A and B then A and B are on opposite sides of C.
Theorem 2.5: Let A be a point on the line determined by the equation ax + by = c, and let B be a point not on that line. Then the points on the line determined by A and B which are on the same side of A as B are on the same side of the line ax + by = c as B, and the points on the other side of A from B on the line determined by A and B are on the other side of the line ax + by = c.
Theorem 2.6: If two lines are parallel, then all of the points on one line lie on the same side of the other line.
Theorem 2.7: Given a points A and B where A which is either on the line or on the same side of the line as B, every point on the line segment between A and B is on the same side of the line as B.
Theorem 2.8: If a line segment contains points on both sides of another line, then the line must intersect the segment somewhere between its endpoints.
Theorem 2.9: Let A, B, and C be three noncolinear points, let D be a point distinct from A on the line determined by A and B, and let E be a point also distinct from A on the line determined by A and C. Then DE is parallel to BC if and only if there is a nonzero real number t such that
and
for the same value of t.
Theorem 2.10: Let A, B, and C be three noncolinear points. If D is on the line through A which is parallel to BC then there is a real number s such that
Theorem 2.11: (The parametric representation of a plane) Let A, B, and C be three noncolinear points. Let D be any point in the plane. Then there are real numbers q, r, and s such that
and
Theorem 2.12: Let A, B, and C be three noncolinear points, and let
where
be a point in the plane. Then D is on the same side of BC as A if and only if q is positive.
Theorem 2.13: (The first Pasch property) Let A, B, and C be three noncolinear points. If a line going through A contains points in the angle between AB and AC, then that line intersects the line segment BC.
Theorem 2.14: (The second Pasch property) Let A, B, and C be three noncolinear points. If a line intersects the line segment AB, then the line will either intersect line segment AC, segment BC, or go through point C.