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9/18/2015 Theorem The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices Diagram When 3 or more lines intersect at one point, they are concurrent. The point at which they intersect is the point of concurrency. The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle. PX , PY , and PZ are concurrent at P. X PA PB PC Y P C Z Since the circumcenter is equidistant from the vertices, you can use the circumcenter as the center of the circle that contains each vertex of the triangle. You say the circle is circumscribed about the triangle. A X Y P B the circumcenter of the following 3 points. Perpendicular bisectors A B Symbols C Z Find Theorem The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle. Diagram Symbols Angle bisectors AP, BP, and CP A are concurrent at P. PX PY PZ X Y P Circumcenter B Z C 1 9/18/2015 If The point of concurrency of the angle bisectors of a triangle is called the incenter of the triangle. P is the center of the circle that is inscribed in the triangle. A X P is the incenter and PY = 2x-4 and PZ = 5x-13, find BP, if PB = 3x. 2 x 4 5 x 13 Y 3x Y P P. PB 9 2x-4 5x-13 B Z 3 x PB 3(3) X P B 9 3x A C Z C 305 #’s 7-10, 15-18, 26-29, 37-38 2