Bisettrice

Bisettrice, known in English as the angle bisector, is a line, ray, or segment that divides a given angle into two congruent angles. There are internal and external bisettrici: the internal bisector lies inside the angle, while the external bisector bisects the angle formed by one side and the extension of the other.

A fundamental property is that any point on the internal angle bisector is equidistant from the two sides of the angle. This distance equivalence characterizes the locus of points on the bisector and underpins many geometric constructions and proofs.

In triangle geometry, the internal angle bisector from a vertex A to the opposite side BC is often studied. If D is the point where the bisector intersects BC, then the Angle Bisector Theorem states AB:AC = BD:DC. The length l_a of the internal angle bisector from A to BC can be expressed as l_a = (2bc cos(A/2))/(b+c) = sqrt(bc[1 − (a^2/(b+c)^2)]), where a = BC, b = AC, and c = AB.

The three internal angle bisectors concur at the incenter, the center of the incircle, which is equidistant from all three sides. External angle bisectors, meanwhile, intersect to form excenters, centers of the excircles tangent to one side and the extensions of the other two.

Construction-wise, a compass-and-straightedge method constructs the internal bisector by marking equal arcs on the two sides from the vertex and drawing a line through the vertex and the intersection of those arcs. The concept of the bisettrice is central to many geometric problems and proofs, including those involving incircles, excenters, and triangle side ratios.