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Special Segments of Triangles. Sections 5.2, 5.3, 5.4. Perpendicular bisector theorem. A point is on the perpendicular bisector if and only if it is equidistant from the endpoints of the segment. Angle Bisector Theorem.

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Special segments of triangles

Special Segments of Triangles

Sections 5.2, 5.3, 5.4


Perpendicular bisector theorem
Perpendicular bisector theorem

  • A point is on the perpendicular bisector if and only if it is equidistant from the

  • endpoints of the segment.


Angle bisector theorem
Angle Bisector Theorem

  • A point is on the bisector of an angle if and only if it is equidistant from the two sides of the angle.


Medians of a triangle
Medians of a triangle

  • A median of a triangle is a segment from a vertex to the midpoint of the

  • opposite side.


Altitudes of a triangle
Altitudes of a triangle

  • An altitude of a triangle is the perpendicular segment from a vertex to the

  • opposite side or to the line that contains the opposite side.


Concurrency
Concurrency

  • The point of intersection of the lines, rays, or segments is called the point of concurrency.


Points of concurrency
Points of concurrency

  • The point of concurrency of the three perpendicular bisectors a triangle is called the circumcenter.

  • The point of concurrency of the three angle bisectors of a triangle is called the incenter.

  • The point of concurrency of the three medians of a triangle is called the centroid.

  • The point of concurrency of the three altitudes of a triangle is called the orthocenter.

  • The incenter and centroid will always be inside the triangle. The circumcenter

  • and orthocenter can be inside, on, or outside the triangle.


What is special about the circumcenter
What is special about the Circumcenter?

  • The perpendicular bisectors of a triangle intersect at a point that is equisdistant

  • from the vertices of the triangle.

PA = PB = PC


What is special about the incenter
What is special about the Incenter?

  • The angle bisectors of a triangle intersect at a point that is equidistant from the

  • sides of the triangle.

PD = PE = PF


What is special about the centroid
What is special about the Centroid?

  • The medians of a triangle intersect at a point that is two thirds of the distance

  • from each vertex to the midpoint of the opposite side.


What is special about the orthocenter
What is special about the Orthocenter?

  • There is nothing special about the point of concurrency of the altitudes of a

  • triangle.