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 line passes through and , as shown in the given diagram.
From and draw and perpendicular to the X-axis and from point draw also perpendicular to the X-axis. Also from draw perpendicular to on .
Now from the given diagram, consider the similar triangles and , and by the definition of a slope we take
Also from the given diagram we have
Putting these all values in the above equation (i) we have
This is the equation of a line passing through two points and . This equation can also have the form
In determinant form, the given equation of a line through two points is
NOTE: There is an alternate way to prove the two points form of the equation of a straight line.
Consider the slope point form of the equation of a line, we have
Since the line is passing through the point in the above equation and the slope of the line is , so equation (i) becomes