 Incenter - Concurrent ∠ Bisectors MathBitsNotebook.com Terms of Use   Contact Person: Donna Roberts  A point of concurrency is the point where three or more lines intersect.

We are now going to take a look at another triangle center called the incenter.

INCENTER - concurrent angle bisectors

The angle bisectors of the angles of a triangle are concurrent (they intersect in one common point). The point of concurrency of the angle bisectors is called the incenter of the triangle. The point of concurrency is always located in the interior of the triangle.

 NOTE: The point of concurrency of the angle bisectors of a triangle (the incenter) is the center of an inscribed circle within the triangle.

An inscribed circle is a circle positioned within a figure such that the circle is tangent to each of the sides of the figure. In this case, the circle is tangent to the sides of the triangle. A circle is tangent to a segment (or line) if it touches the segment only once, but does not cross the segment. Since radii in a circle are of equal length, the incenter is equidistant from the sides of the triangle.      Locate the incenter through construction: We have seen how to construct angle bisectors of a triangle. Simply construct the angle bisectors of the three angles of the triangle. The point where the angle bisectors intersect is the incenter. Actually, finding the intersection of only 2 angle bisectors will find the incenter. Finding the third angle bisector, however, will ensure more accuracy of the find. Construct a circle inscribed in a triangle. When we circumscribed a circle about a triangle, we determined the radius of the circle by measuring the distance from the center to a vertex. When constructing an inscribed triangle, we can "eyeball" the radius of our circle, but we have no actual length to measure. We need a length, since "eyeballing" is not sufficient. To determine the length, we will construct a perpendicular to one side from the center point to locate the radius. Theorem: The radius of a circle drawn to the point of tangency of a tangent line is perpendicular to the tangent. Steps: 1. locate the incenter by constructing the angle bisectors of at least two angles of the triangle. 2. construct a perpendicular from the incenter to one side of the triangle to locate the exact radius. 3. place compass point at the incenter and measure from the center to the point where the perpendicular crosses the side of the triangle (the radius of the circle). 4. draw the circle.  