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.

Centroid Animation

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?


Orthocenter Animation


Why does the Othocenter move the way it does?
(It may help to consider how the Orthocenter is constructed.)


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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.

Circumcenter 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:



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?


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?


Center Animation

Try adjusting point X around the circle:

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