The perpendicular bisectors of the sides of a triangle intersect at a single point called the circumcenter. The circumcenter is equidistant from the triangle's vertices and can be used to locate the circle that passes through all three vertices (the circumscribed circle). Similarly, the angle bisectors of a triangle intersect at the incenter, which is equidistant from the triangle's sides and located inside the triangle. The document provides examples of finding the circumcenter and using the incenter to solve a math problem involving a triangle.

2.  When three or more lines intersect at one point,
they are concurrent.
 The point at which they intersect is the point of
concurrency.

3.  The perpendicular bisectors of the sides of a triangle
are concurrent at a point equidistant from the
VERTICES.

4.  The point of concurrency of the perpendicular
bisectors of a triangle is called the circumcenter
of the triangle.
 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 would say the circle is circumscribed about the
triangle.

5.  The circumcenter of a triangle can be inside, on,
or outside the triangle.

6.  What are the coordinates of the circumcenter of the triangle
with vertices P(0 , 6), O(0 , 0), and S(4 , 0)?
 Find the intersection point of the triangle’s perpendicular
bisectors. It is easiest to find the perpendicular bisectors of
segment PO and segment OS.

7.  The bisectors of the angles of a triangle are
concurrent at a point equidistant from the SIDES of
the triangle.

8.  The point of concurrency of the angle bisectors
of a triangle is called the incenter of the
triangle.
 The incenter is ALWAYS inside the triangle.

9.  GE = 2x – 7 and GF = x + 4. What is GD?
 G is the incenter of the triangle.
GE = GF
2x – 7 = x + 4
x = 11
GF = 11 + 4
= 15
Since GF = GD, GD = 15.

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