Bisectors of Triangles

Perpendicular Bisector

A perpendicular bisector of a side of a Triangle is a Line, segment, or ray that is perpendicular to that side at its midpoint.

• Perpendicular Bisector Theorem: Any Point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.
• Converse: If a Point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector.

Circumcenter

The three perpendicular bisectors of a Triangle are concurrent (meet at one Point). This Point is called the circumcenter.

• The circumcenter is equidistant from all three vertices.
• It is the center of the circumscribed Circle (the Circle passing through all three vertices).
• In an acute Triangle, the circumcenter is inside the Triangle.
• In a right Triangle, the circumcenter is on the hypotenuse (its midpoint).
• In an obtuse Triangle, the circumcenter is outside the Triangle.

Angle Bisector

An Angle bisector of a Triangle divides an Angle into two congruent angles.

• Angle Bisector Theorem: Any Point on the Angle bisector is equidistant from the two sides of the Angle.

Incenter

The three Angle bisectors of a Triangle are concurrent. This Point is called the incenter.

• The incenter is equidistant from all three sides.
• It is the center of the inscribed Circle (the Circle that fits inside the Triangle and touches all three sides).
• The incenter is always inside the Triangle.