No, any two different lines with the same slope will not intersect.
Practice makes perfect
Fisher states that any two lines will have a point of intersection. We need to either prove this is true or give a counter example. Let's start with an arbitrary line. We can determine its slope by taking the quotient of the run over the rise between two points.
In this case the slope is equal to 1. This means that the line rises one unit for every horizontal unit it moves in the positive direction. We can add another line above this one with a less steep slope, but slanted upwards as well. We see that the first line will intersect the new line at some point. This is because it rises faster than the new one.
If we add a new line with a steeper slope, the extension of this new line will intersect the original line at some point in the negative x values. This is because it would be decreasing faster in negative values than the first line.
Now we are just left to consider what happens if both lines have the same slope. In this case, no line will intersect the other. For every step to the left or right each line would decrease or rise by the same amount. They will not get any closer! Therefore, they will not intersect at any point.
We have proven that not all lines have a point of intersection. Only lines that have different slopes will intersect.
