an angle bisector

Open a pair of scissors and imagine a line running straight down the middle of the V, splitting the opening into two equal angles. That dividing ray is the bisector of the angle.

Precisely, the bisector of angle ABC is the ray BD, with D in the interior of the angle, that divides it into two equal angles: m(angle ABD) = m(angle DBC). By the angle addition postulate each equals half the original, so m(angle ABD) = m(angle DBC) = (1/2)·m(angle ABC). Just as a segment has exactly one midpoint, an angle has exactly one bisector. The bisector can be built with compass and straightedge: from the vertex swing an arc cutting both sides, then from those two crossings swing equal arcs that meet inside — the line from the vertex through that meeting point is the bisector.

The angle bisector earns its keep through a beautiful property: every point on the bisector is equidistant from the two sides of the angle (measuring the perpendicular distance to each side). This makes bisectors the path to the incentre of a triangle — the three angle bisectors meet at one point, the centre of the inscribed circle. The angle bisector is the angle-world twin of the midpoint, and it should be kept distinct from the perpendicular bisector of a segment, which is a different construction solving a different equidistance problem.