Introduction to Bisectors

A bisector divides a segment into two equal parts. In triangles, bisectors can occur from angles or sides, creating interesting properties and precise divisions within the geometric shape.

An angle bisector in a triangle splits the angle into two equal parts. It has a unique trait: each point on the bisector is equidistant from the triangle's two sides forming the angle.

Where all three angle bisectors intersect, the incenter is formed. Surprisingly, it's the center of the inscribed circle (incircle), touching each side of the triangle at exactly one point.

Perpendicular bisectors are not only equidistant from endpoints of a side but meet at the circumcenter. This point is the center of the circumscribed circle (circumcircle) around the triangle.

In equilateral triangles, all bisectors from angles and sides coincide, intersecting at the center. For isosceles triangles, the altitude, median, angle bisector, and perpendicular bisector are the same for the base.

The concurrency of angle bisectors at the incenter and perpendicular bisectors at the circumcenter are examples of Ceva's and Euler's lines, fundamental concepts in triangle geometry.

Bisectors have practical applications, such as in trilateration, the process used by GPS technology to determine precise locations by measuring distances to multiple points.

Ancient Egyptians used bisectors to align the pyramids accurately with the cardinal points, demonstrating their advanced understanding of geometry.