1.5 Special Points in TrianglesObjectives: Discover points of concurrency in triangles.Draw the inscribed and circumscribed circles of triangles.Warm-Up:

How many times can you subtract the number 5 from 25?Once. After the first calculation you will be subtracting 5 from 20, then 5 from 15, and so on.

If you divide thirty by a half and add ten, what is the answer?

70. When dividing by fractions, you must invert and multiply.

Vocabulary:Inscribed Circle:

A circle in the inside of a triangle that touches each side at one point.

(a circle is inscribed in a polygon if each side of the polygon is tangent to the circle)

Vocabulary:Circumscribed Circle:

A circle that is drawn around the outside of a triangle and contains all three vertices.

(a circle is circumscribed about a polygon if each vertex of the polygon lies on the circle)

Vocabulary:Concurrent:

Literally “running together” of three or more lines intersecting at a single point.

Vocabulary:Incenter:

The center of a inscribed circle; the point where the three angle bisectors intersect.

(It is equidistant from the three sides of the triangle).

Vocabulary:Circumcenter:The center of a circumscribed circle where the three perpendicular bisectors of the sides of a triangle intersect.

(It is equidistant from the three vertices of the triangle).

Example 1:Label the inscribed circle, circumscribed circle, the incenter, the circumcenter, and points of concurrency in the following figures.

inscribed circlecircumscribed circle

incentercircumcenter

points of concurrency

Example 2:Find the perpendicular bisector of each side of XYZ .

Z

Y

X

Example 3:Find the angle bisectors of each angle of LMN .

L

M

N

Example 4:Find the circumscribed circle of JKL.

J

K

L

To find identify the perpendicular bisectors of the triangles sides.

Example 5:Find the inscribed circle of MNO.

M

N

O

To find identify the angle bisectors of the triangle.

Example 6:Find the circumscribed circle of .

A

B C

To find identify the perpendicular bisectors of the triangles sides.

Example 7:Find the inscribed circle of WXY.

W

X

Y

To find identify the angle bisectors of the triangle.