Analytic Foundations of Geometry

1. Equations of Lines

Robert S. Wilson

Theorems 1.1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

Theorem 1.1: If   (x1, y1)  and  (x2, y2)  are any two points on the line whose equation is  y = mx + b,  then the slope between them is  m.1

Theorem 1.2: A vertical line has infinite slope2. A line is not vertical if and only if its equation can be written in the form

y = mx + b

Theorem 1.3: There is a uniquely determined line going through a given point with a given slope. If the given point is  (x1, y1),  and the line is vertical3, then the equation is

x = x1

If the line is not vertical, then the equation can be written as

y - y1 = m(x - x1)

which can be simplified to

y = mx + b

where

b = y1 - mx1

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Theorem 1.4: Two points uniquely determine a line. If the points are  (x1, y1)  and  (x2, y2)  then the equation is

x = x1

if the two points are vertical, and if not then it is of the form

y = mx + b

where

and

Theorem 1.5: If   (x1, y1)   and   (x 2, y2)  are points on the line whose equation is  y = mx + b   then the distance between them isT1

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Theorem 1.6: If two distinct lines are parallel, then they do not meet. If they are not parallel, then they meet at a uniquely determined point. If one of the lines is vertical, its equation will be of the form

x = a

If the other one is not vertical, it will have an equation of the form

y = mx + b

and the point of intersection will be

(a, ma + b)

If neither of the lines are vertical, they will have equations of the form

The coordinates of the point of intersection 4 are

and

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Theorem 1.7: If we are given a line and a point not on the line, then there is a unique line, going through a given point, parallel to the given line.5

Theorem 1.8: If we are given a point and a line, there is a unique line, through the point, perpendicular to the given line.

Theorem 1.9: Given a line whose equation is

y = mx + b

and a point whose coordinates are

(x1, y1)

the equation of the unique line through the point, perpendicular to the line, is

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Theorem 1.10: Given a line whose equation is

y = mx + b

and a point whose coordinates are

(x1, y1)

the coordinates of the foot of the point in the line are

Theorem 1.11: The perpendicular distance from the point

(x1, y1)

to the line

y = mx + b

is

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Theorem 1.12: Parallel lines stay the same distance apart. If the equations of the lines are

then the perpendicular distance from any point on one line to the other line isT2

If the lines are vertical, then their equations are

x = a1   and   x = a2

and the distance between the lines is

|a1 - a2|

Theorem 1.13: A line is parallel to one of two parallel lines if and only if it is parallel to the other.

Theorem 1.14: If a line is perpendicular to one of two parallel lines then it is perpendicular to the other.

Theorem 1.15: If two lines are perpendicular to the same line then they are parallel.

Theorem 1.16: If a line is parallel to one of two perpendicular lines, it is perpendicular to the other.

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2. Parametric Equations