Special lines in Triangles and their points of concurrency Perpendicular bisector of a triangle: is perpendicular to and intersects the side of a triangle at its midpoint Circumcenter: the point of concurrency for perpendicular bisectors Center of circumscribed circle Center of circumscribed circle Equal distances from the circumcenter to the vertices Equal distances from the circumcenter to the vertices

Perpendicular Bisectors and the Circumcenter

Special Lines and Points… Angle Bisectors of a Triangle: Lines that cut the angles of a triangle into two congruent angles. Incenter: The point of concurrency for the angle bisectors. The center of the inscribed circle The center of the inscribed circle The incenter is equidistant from the sides of the triangle The incenter is equidistant from the sides of the triangle

Angle bisectors and the Centroid

Special Lines and Points… Medians: A line drawn from the vertex of a triangle to the midpoint of the opposite side. Centroid: Point of Concurrency for the medians The distance from the vertex to the centroid is 2/3 the distance to the side of the triangle The distance from the vertex to the centroid is 2/3 the distance to the side of the triangle The shorter segment of each median then is 1/3 the distance from the vertex to the midpoint The shorter segment of each median then is 1/3 the distance from the vertex to the midpoint The shorter segment of each median is also ½ the larger segment The shorter segment of each median is also ½ the larger segment

Medians and the Centroid

Special Lines and Points… Altitudes: A line drawn from the vertex of a triangle perpendicular to the opposite side. Orthocenter: Point of Concurrency for the altitudes The orthocenter may be inside, outside, or on the triangle The orthocenter may be inside, outside, or on the triangle

Altitudes and the Orthocenter