Geometry Chapter 5.2 Bisectors of a Triangle
I) Vocabulary:
A) A Perpendicular bisector of a triangle - is a line (or ray or segment) that is perpendicular to a side of the triangle at the midpoint of the side.
B) Concurrent lines - when three or more lines (or rays or segments) intersect in the same point.
C) Point of concurrency - the point of intersection of the concurrent lines.
D) Circumcenter of the triangle - the point of concurrency of the perpendicular bisectors of a triangle.
E) Angle bisector of a triangle - is a bisector of an angle of the triangle.
F) Incenter of the triangle - the point of concurrency of the angle bisectors of a triangle.
II) Theorems:
A) 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.
Example: Given ΔABC with point P being the circumcenter of the triangle, we can conclude that
PA = PB = PC.
Using point P as the center of the circle and PA, PB, or PC as a radius, the circleis circumscribed about the triangle.
B) 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.
Example: Given ΔABC with point P being the incenter of the triangle, we can conclude that
PD = PE = PF and PD ll BC, PE ll AC and PF ll AB.
Using point P as the center of the circle and PD, PE, or PF as a radius, the circle is inscribed within the triangle.
