Envision Geometry - Topic 5

Perpendicular Bisector Theorem

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

Concurrency

When 3 or more lines intersect at one point

circumcenter

the point of concurrency of the perpendicular bisectors of the sides of the triangle. equal distance from all the vertices of a triangle.

circumscribed circle

the circle that contains the vertices of an inscribed polygon

incenter

the point of concurrency of the angle bisectors of a triangle. equal distance from all the sides of the triangle.

inscribed circle

A circle that is tangent to all three sides of the triangle

median

a segment from a vertex to the midpoint of the opposite side

altitude

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

centroid

The point of concurrency of the medians of a triangle

Hinge Theorem

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.

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

orthocenter

the point of concurrency of the three altitudes of a triangle

The vertices

The circumcenter is equidistant from the

The sides

The incenter is equidistant from the

median

a segment from a vertex to the midpoint of the opposite side

altitude

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

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.

Converse of the Angle Bisector Theorem

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

Theorem 5-9

If two sides of the same triangle are not congruent, then the larger angle is opposite the longer side.

Converse of Theorem 5-9

If two angles of the same triangle are not congruent, then the longer side is opposite the larger angle.

Converse of the Hinge Theorem

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.