The proofs of these formulas can be found in the Analytic Foundations of Geometry.

1. The equation of the line that joins the points (x_{1},
y_{1}) and (x_{2}, y_{2}) is y = mx + b where
m =

Once you know the slope, one can use the point-slope form of the equation to find the y intercept, but there are formulas.

2. The coordinates of the point where the lines

meet are

and

3. The foot of the point (x_{1}, y_{1}) in the
line y = mx + b has coordinates

4. The distance from the point (x_{1}, y_{1}) to
its foot in the line y = mx + b is

5. If (x_{1}, y_{1}) and (x_{2},
y_{2}) are two points on the line y = mx + b, then the
distance between them is.

6. The coordinates of the points where the line y = mx + b meets
the circle (x- x_{1})^{2} + (y -
y_{1})^{2} = r^{2} are

and

or

and