The equation of a non-vertical line passing through two points and is given by

To prove this equation let

be any point on the given line

. Also this passing thorough

and

as shown in the given diagram.

Form

and

draw

and

perpendicular on X-axis and from point

draw

also perpendicular on X-axis. Also from

draw perpendicular

on

.

Now from the given diagram, consider the similar triangles

and

, by the definition of slope we take

Also from the given diagram we have

Putting these all values in above equation (i) we have

Which is the equation of line through two points and . This equation can also have the from

In determinant from, the given equation of a line through two points is

NOTE: There is an alternate way to prove two points form of equation of a straight line.

Consider the slope point form of equation of a line, we have

Since the line passing through the point

, above equation and we have slope of line is

so equation (i) becomes

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