5.2: Bisectors of a Triangle GEOMETRY: Chapter 5 5.2: Bisectors of a Triangle

Concurrency When three or more lines, rays, or segments intersect in the same point, they are called concurrent lines, rays, or segments. The point of intersection of the lines, rays, or segments is called the point of concurrency.

Theorem 5.6: Concurrency of Perpendicular Bisectors of a Triangle The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. Images taken from: Geometry. McDougal Littell: Boston, 2007. P. 305.

Ex. 1: Each of the three forest ranger stations is the same distance from the main office. Describe how to find the location of the office. Answer: Draw the perpendicular bisectors of the three sides. Those three lines meet at a point, and that point is the location of the office.

Theorem 5.7: Concurrency of Angle Bisectors of a Triangle. The angle bisectors of a triangle intersect at a point that is equidistant from the sides of a triangle. Images taken from: Geometry. McDougal Littell: Boston, 2007. P. 312.

The circle is inscribed within the triangle. The point of concurrency of the three angle bisectors of a triangle is called the incenter of the triangle. The incenter always lies inside the triangle. A circle can be drawn using P as the center and the distance to one aside as the radius will just touch the other two sides. The circle is inscribed within the triangle. Images taken from: Geometry. McDougal Littell: Boston, 2007. P. 312.

Circumcenter—The point of concurrency of the three perpendicular bisectors of a triangle is called the circumcenter of the triangle. The circumcenter P is equidistant from the three vertices, so P is the center of a circle that passes through all three vertices. Images taken from: Geometry. McDougal Littell: Boston, 2007. P. 306.

Ex. 2: In the diagram, G is the incenter of triangle RST. Find GW. Images taken from: Geometry. McDougal Littell: Boston, 2007. P. 312.

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