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Parallel and Perpendicular Lines
A
B
C
D
E
. . . .

Perpendicular Lines

What is the shortest distance from a point to a line? The next theorem answers this question.

Theorem. First Minimum Theorem
The shortest segment joining a point to a line is the perpendicular segment.

Given: line l and an external point P
PA ¯ l at point A.
B is any other point on line l.
Prove: PA<PB

Proof: