Honors Geometry 9/10 Chapter 5 Relationships in Triangles

Perpendicular Bisector Theorem

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment

Converse of the Perpendicular Bisector Theorem

If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment

Angle Bisector Theorem

If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle

Converse of the Angle Bisector Theorem

If a point is equidistant from two sides of an angle, then it is on the angle bisector

Concurrency of Perpendicular Bisectors Theorem

The perpendicular bisectors of the sides of a triangle are concurrency at a point equidistant from the vertices

Concurrent

When three or more lines intersect at one point

Point of Concurrency

The point at which concurrent lines intersect

Concurrency of Angle Bisectors Theorem

The angle bisectors of a triangle are concurrent at a point equidistant from the sides of the triangle

Inscribed Circle of a Triangle

the circle that intersects each side of a triangle at exactly one point and has no points outside of the triangle

Incenter

the point of concurrency of the angle bisectors of a triangle

The incenter is equidistant from…

the sides of the triangle

Circumscribed circle

The circle that contains all three vertices of a triangle

Circumcenter

the point of concurrency of the perpendicular bisectors of a triangle

The circumcenter is equidistant from…

the vertices of the triangle

Median of a triangle

a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side

Concurrency of Medians Theorem

The medians of a triangle are concurrent at a point that is 2/3 the distance from each vertex to the midpoint of the opposite side

Concurrency of Altitudes Theorem

The lines that contain the altitudes of a triangle are concurrent

Altitude of a triangle

a perpendicular segment from a vertex to the line containing the opposite side

Centroid

The point of concurrency of the medians of a triangle

The centroid is…

2/3 of the distance from each vertex to the midpoint of the opposite side

Orthocenter

The point of concurrency of the altitudes of a triangle