**Theorem
1.10**: Given a line
whose equation is

and a point whose coordinates are

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

**Proof**: By Theorem 1.6,
the coordinates of the
point of intersection of
the perpendicular
line through
* * (*x*_{1}, *y*_{1}) * * with the line * * *y* = *mx* + *b* * * are

and

where

by Theorem 1.9,

and

We substitute these in and get

by Theorem 1.6. Multiply top
and bottom by *m* to clear denominators

If we substitute this into *y* = *mx* + *b*, we get

We need common denominators.

Remove parentheses and add the fractions

or