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