Warm-Up

Three or more lines that intersect at the same point are called

concurrent lines

. The point of intersection is called the

point of concurrency

.

C A E G F B D

Example 1

Are the lines represented by the equations below concurrent? If so, find the point of concurrency.

Pick 2 equations and solve them for x & y

x

+

y

= 7

x

+ 2

y

= 10 Plug the values into all 3 equations and see if they make true statements

x

-

y

= 1 x=4 y=3 Yes

5.2-5.4: Points of Concurrency

Objectives: 1.

To define various points of concurrency 2.

To discover, use, and prove various theorems about points of concurrency

Intersecting Medians Activity

The centroid of a triangle divides each median into two parts. Click the button below to investigate the relationship of the 2 parts.

Concurrency of Medians Theorem

The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side.

Centroid

The three medians of a triangle are concurrent. The point of concurrency is an interior point called the

centroid

. It is the balancing point or center of gravity of the triangle.

Example 2

In Δ

RST

,

Q QW

and is the centroid and

SW

.

SQ

= 8. Find QW = 4 SQ = 12

A

Others Points of Concurrency

Since a triangle has 3 sides, it seems obvious that a triangle should have 3 perpendicular bisectors, 3 angle bisectors, and 3 altitudes. But are these various segments concurrent?

A B A C B C B C

A

Others Points of Concurrency

In this activity, we will use patty paper to investigate other possible points of concurrency, and then, hopefully, something magical will happen… A B A C B C B C

Circumcenter

Concurrency of Perpendicular Bisectors of a Triangle Theorem

The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.

Circumcenter

The point of concurrency of the three perpendicular bisectors of a triangle is called the

circumcenter

of the triangle.

In each diagram, the circle

circumscribes

the triangle.

Explore

Explore the perpendicular bisectors of a triangle and its circumcenter by clicking the button below

Incenter

Concurrency of Angle Bisectors of a Triangle Theorem

The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle.

Incenter

The point of concurrency of the three angle bisectors of a triangle is called the

incenter

of the triangle.

In the diagram, the circle is

inscribed

within the triangle.

Explore

Explore the angle bisectors of a triangle and its incenter by clicking the button below

Orthocenter

Concurrency of Altitudes of a Triangle Theorem

The lines containing the altitudes of a triangle are concurrent.

G

Orthocenter

The point of concurrency of all three altitudes of a triangle is called the

orthocenter

of the triangle.

The orthocenter, P, can be inside, on, or outside of a triangle depending on whether it is acute, right, or obtuse, respectively.

Explore

• Explore the altitudes of a triangle and its orthocenter by clicking the button below.

Example 3

Is it possible for any of the points of concurrency to coincide? In other words, is there a triangle for which any of the points of concurrency are the same.

Record your thoughts/predictions in your notebook

Example 4

Is it possible for any of the points of concurrency to be collinear?

Euler Line

The

Euler Line

is the line that contains the

o

rthocenter,

c

entroid, and the

c

ircumcenter of a triangle.

A C Orthocenter Centroid Ci rcumcenter B

Explore

Click the button below to explore the Euler Line

Calculate in your notebook

Calculate in your notebook