# Assignment 4:

Concurrencies of a Triangle

## Relative Locations of Triangle
Centers

**A **__Centroid__ is
formed in a triangle by the intersection of segments from the
MIDPOINT of the SIDES to the OPPOSITE VERTEX, like this:
**Think about how the Centroid
is constructed.**

Do you think the Centroid is always on the Interior of the Triangle?

How can you test your thoughts?

**Take a look at an animated
version of a Triangle with its Centroid.**
Can you explain why the Position of the Centroid behaves
the way it does as we change the Triangle?

**An **__Orthocenter__
is formed in a triangle by the intersection of the ALTITUDES:
**We can construct the Orthocenter
by making lines Perpendicular from each side to the Opposite vertex,
like this:**

**Do you think the Orthocenter
is always inside the Triangle?**
###

Why does the Othocenter move the way it does?

(It may help to consider how the Orthocenter is constructed.)
###

### Need Help? Click below.

###

### The __Circumcenter__
is formed by the intersections of the

PERPENDICULAR BISECTOR to Each Side of the Triangle, like this:

###

### Do you think the Circumcenter
can ever lie on the Exterior of the Triangle?

Take a look at the animation.
Did the Circumcenter behave the
way you expected it to?
Can you explain why it behaves the it does?
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# We can construct any Triangle
within a Circle:

###

### n

### What do you think might happen
to the __Centroid__, the __Orthocenter__,
and the __Circumcenter__ as we adjust the Triangle within
the Circle? That is, if we move any vertex of the Triangle around
the Perimeter of the Circle, how will the Centers behave - will
they lie inside or outside the Triangle?

### n

### Take a look at the
animation below.

### It shows a Triangle
inscribed in a Circle and

There are several points labled: A, B, & C.

Based on what you saw in the discussions above,can you determine
which Triangle Centers are represented by A, B, & C?

# Try adjusting point
X around the circle:

# What observations can
you make about the relationships between the Triangle, the Centers,
and the Circle?