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 (x1, y1) 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
