5.2 B ISECTORS OF A T RIANGLE We have learned about the perpendicular bisector of a segment and the bisector of an angle. Now we will learn about the special cases in which the segments and angles of a triangle are bisected. Perpendicular Bisector of a Triangle: a line, segment or ray that is perpendicular to a side of the triangle at the midpoint of the side. Work Together Fold the paper to create the perpendicular bisectors of each side of an acute triangle. What do you notice ? Repeat the same steps for the right triangle. What do you notice? Draw an obtuse triangle in the middle of a piece of paper. Fold the paper to create the perpendicular bisectors. What do you notice? Make a conjecture about the perpendicular bisectors of the sides of a triangle.

Concurrent Lines (segments or rays) : Three or more lines intersecting at a point. Point of Concurrency: The point at which the lines intersect. The three perpendicular bisectors of a triangle are concurrent. The point of concurrency will be inside the triangle if it is an acute triangle, on the triangle if it is a right triangle and outside the triangle if it is an obtuse triangle. The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle. (This has a special property shown in theorem 5.5.) Theorem 5.5 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. PA = PB = PC A C B P

U SING A NGLE B ISECTORS OF A T RIANGLE Angle Bisector of a Triangle: a bisector of an angle of the triangle. Work Together Fold the paper to create the angle bisectors of each angle of an acute triangle. What do you notice about the angle bisectors? Repeat the same steps for the right and obtuse triangles. Does your observation in the first part still hold true? Make a conjecture about the angle bisectors of a triangle. The three angle bisectors are concurrent. The point of concurrency is called the incenter of the triangle. (This has a special property shown in theorem 5.6.) Theorem 5.6 Concurrency of Angle Bisectors of a Triangle: The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. PD = PE = PF A B C F E D P