Geometry Final Theorms

Perpendicular Bisector Theorem

In a plane, if a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment

Converse of Perpendicular Bisector Theorem

In a plane, if a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment.

Angle Bisector Theorem

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

Converse of the Angle Bisector Theorem

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

Circumcenter Theorem

The circumcenter of a triangle is equidistant from the vertices of the triangle

Incenter Theorem

The incenter of a triangle is equidistant from the sides of the triangle

Incenter

the point where the three angle bisectors of a triangle intersect

Circumcenter

the point where the three perpendicular bisectors of a triangle intersect

Centroid Theorem

The centroid of a triangle is two-thirds of the distance from each vertex to the midpoint of the opposite side

Centroid

the point where the three medians (segment from vertex to the midpoint of the opposite side) of a triangle intersect

Orthocenter

The lines containing the altitudes of a triangle are concurrent. This point of concurrency is the orthocenter of the triangle

Midsegment of a triangle

segment that connects the midpoints of two sides of the triangle. Each triangle has three

Triangle Midsegment Theorem

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.

How to Write an indirect Proof

1. Identify the statement you want to prove. Assume temporarily that this statement is false by assuming that its opposite is true

2. Reason logically until you reach a contradiction

3. Point out that the desired conclusion must be true because that contradictions proves the temporary assumption false

Triangle Longer Side Theorem

If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side.

Triangle Larger Angle Theorem

If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.

Triangle Inequality Theorm

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

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.

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