Parallel and perpendicular lines

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A straight-line can be written in the form math-container{y-yindex{1}=m(x-xindex{1})} where math-container{(xindex{1},yindex{1})} is a point on the line and math-container{m} is the gradient. The gradient of the straight line through two points math-container{(xindex{1},yindex{1})} and (math-container{xindex{2},yindex{2})} is math-container{m=frac{yindex{2}-yindex{1}|xindex{2}-xindex{1}}} The distance between two points math-container{(xindex{1},yindex{1})} and (math-container{xindex{2},yindex{2})} is given by the formula math-container{sqrt{(xindex{1}-xindex{2})power{2}+(yindex{1}-yindex{2})power{2}}}
The coordinates of the midpoint of the line joining math-container{(xindex{1},yindex{1})} and (math-container{xindex{2},yindex{2})} are given by the formula math-container{(frac{xindex{1}+xindex{2}|2},frac{yindex{1}+yindex{2}|2} )} Two lines are described by the equations math-container{yindex{1}=mindex{1}x+cindex{1}} and math-container{yindex{2}=mindex{2}x+cindex{2}} If math-container{mindex{1}=mindex{2}} the two lines are parallel If math-container{mindex{1}× mindex{2}=-1} ( i.e math-container{mindex{1}} is the negative reciprocal of math-container{mindex{2}}) the two lines are perpendicular.