Perpendicular bisectors and equidistance

The perpendicular bisector passes through the midpoint and meets the segment at a right angle. Its points are equally distant from the endpoints: the two right triangles have equal half-bases and a common perpendicular side. Find the midpoint and segment gradient. Use the negative reciprocal for the bisector through the midpoint. For a horizontal segment the bisector is vertical, and conversely. Combine with any second line to locate an equidistant point. For A=(0,0),B=(4,2), M=(2,1), m_AB=1/2. Thus the bisector is y-1=-2(x-2), or y=-2x+5. Check: a perpendicular line through an endpoint is not the perpendicular bisector. Both the midpoint and right-angle conditions are essential.