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Larson Geometry Reference – Chapter 6
Definitions
Term/Concept
*Altitude
Centroid
Circumcenter
Description
A line segment from a vertex of a triangle to the side opposite the vertex, perpendicular to that
side.
The point of concurrency of the medians of a triangle.
The point of concurrency of three perpendicular bisectors of a triangle.
B
G is the circumcenter of ABC.
Therefore AG  BG  CG because they are radii
Of the circle going through the vertices A, B, and C.
Concurrent
*Equidistant
Incenter
Inequality
*Median of a Triangle
*Midsegment of a Triangle
Orthocenter of a Triangle
*Perpendicular Bisector
*Point of Concurrency
E
D
G
A
F
C
When three or more lines, rays, or segments intersecting at the same point they are called
concurrent. That point is the point of concurrency.
The property of two or more objects being the same distance from an another object.
The point of concurrency of the angle bisectors of a triangle.
For any real numbers a and b, a > b if and only if there is a positive number c such that a = b +
c.
A line segment that connects a vertex of a triangle to the midpoint of the side opposite that
vertex.
The segment that connects the midpoints of two sides of a triangle.
The point at which the lines containing the three altitudes of a triangle intersect. The point of
concurrency of the three altitudes of a triangle.
A segment, ray, line, or plane that is perpendicular to a segment at its midpoint.
The point at which concurrent lines rays or segments intersect.
Larson Geometry Reference – Chapter 6
Theorems
Theorems
*Angle Bisector Theorem
*Centroid Theorem
Circumcenter Theorem
Concurrency of Altitudes of a
Triangle
Concurrency of Angle
Bisectors of a Triangle
Concurrency of Medians of a
Triangle
Concurrency of Perpendicular
Bisectors of a Triangle
Converse of the Angle
Bisector Theorem
Converse of the Perpendicular
Bisector Theorem
Converse of the Hinge
Theorem
Hinge Theorem
*Incenter Theorem
Orthocenter Theorem
*Perpendicular Bisector
Theorem
Triangle Longer Side
Theorem
Triangle Larger Angle
Theorem
*Triangle Inequality Theorem
*Triangle Midsegment
Theorem
Any point on the bisector of an angle is equidistant from the sides of the angle.
The centroid of a triangle is two-thirds the distance from the vertex to the midpoint of the opposite
side along a median.
The circumcenter of a triangle is equidistant from the vertices of the triangle.
The lines containing the altitudes of a triangle are concurrent.
The angle bisectors of a triangle intersect at ta point that is equidistant from the sides of the triangle.
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.
The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of
the triangle.
Any point on or in the interior of an angle and equidistant from the sides of an angle lies on the
bisector of the angle.
In a plane, any point equidistant from the endpoints of a segment lies on the perpendicular bisector
of the segment.
If two sides of one triangle are congruent to two sides of another triangle, and the third side of the
first is longer than the third side of the second, then the included angle of the first is larger than the
included angle of the second.
If two sides of one triangle are congruent to two sides of another triangle, and the included angle of
the first is larger than the included angle of the second, then the third side of the first is longer than
the third side of the second.
The incenter of a triangle is equidistant from the sides of the triangle.
The lines containing the altitudes of a triangle are concurrent.
In a plane, any point on the perpendicular bisector of a segment is equidistant from the endpoints of
the segment.
If one side of a triangle is longer than another side, then the angle opposite the longer side has a
greater measure than the angle opposite the shorter side.
If one angle of a triangle has a greater measure than another angle, then the side opposite the greater
angle is longer than the side opposite the lesser angle.
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is
half as long as that side.