Honors geometry chapter 5

Very sad

equidistant

a a point is this from two figures when the point is the same distance from each figure

Perpendicular bisector theorem

if a point lies on the perpendicular bisector of a 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 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 the two sides of the angle

Converse of the angle bisector theorem

if a a point lies in the interior of 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 equidistance from the vertices of the triangle

Concurrent lines

When three or more lines, rays, or segments intersect the same point,.

Point of concurrency

the point of intersection of the lines, rays, or segments

Circumcenter

In a triangle, the three perpendicular bisectors are concurrent. The point of concurrency is this.

Incenter theorem

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

Median of a triangle

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

Centroid

The three medians of a triangle are concurrent. The point of concurrency is this.

Centroid theorem

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

Altitude of a triangle

The perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side.

Orthocenter

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

Incenter

In a triangle, the three angle bisectors are concurrent. The point of concurrency is this.

Mid segment of a triangle

A segment that connects the midpoints of two sides of the triangle. Every triangle has 3 midsegments which form the midsegment triangle.

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.

Indirect proof

You start by making the temporary assumption that the desired conclusion is false. By then showing that this assumption leads to a logical impossibility, you prove the statement true.

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 he shorter side.

Triangle larger angle theorem

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

Triangle inequality theorem

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 triangle is larger than the included angle of the second, then the third side of the first triangle 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 then the third side of the second, then the included angle of the first is larger than the included angle of the second.