Assignment 4

Exploring the centroid, orthocenter,

circumcenter, and incenter

By Carly Coffman


Centroid

The CENTROID (G) of a triangle is defined as the common intersection of the three medians. A median of a triangle is the segment from a vertex to the midpoint of the opposite side.

 

We can explore the centroid by using Geometer's Sketch Pad. Click on the link below to explore and make observations about the centroid.

Centroid

?1) Are there any cases where the centroid is outside the triangle? If so, when did it happen?

?2) What happens to the circumcenter when there is a degenerate triangle (when the triangle becomes a line)?

 

Now, with Geometer's Sketch Pad, (GSP), create the centroid of a triangle. Here are the steps and you can use the link below to get back to GSP.

GSP

1) With the mouse, highlight the point tool from the toolbox. (If you do not have a toolbox appear on your sketch go to Display on the top toolbar and go down to Show toolbox.)

2) Create three points on your sketch by clicking with the mouse.

3) Drag your mouse back to the toolbox and click on the arrow (this allows you to highlight).

4) Highlight the three points

5) Go to the top toolbar, select Construct, and go down to Segments

6) Click off the triangle to de-highlight all objects

7) Create the midpoint of each side of the triangle by selecting the three sides (make sure nothing else on your sketch is selected), then go up to the toolbar, highlight Construct and then go down to midpoint.

8) Repeat step #6, then highlight one vertex and the opposite side's midpoint, and repeat steps #5 and #6

9) Highlight another vertex and the opposite side's midpoint and repeat steps #5 and #6

10) Highlight the last vertex and the opposite side's midpoint and repeat steps #5 and #6

11) Highlight the intersection of the three lines you just created. Congratulations, you just created your own centroid!

(Do not save when exiting GSP)


Orthocenter

The ORTHOCENTER (H) of a triangle is the common intersection of the three lines containing the altitudes. An altitude is a perpendicular segment from a vertex to the line of the opposite side.

We can explore the orthocenter by using Geometer's Sketch Pad. Click on the link below to explore and make observations about the orthocenter.

Orthocenter

?3) Were there any cases where the orthocenter was outside the triangle? If so, when did it happen?

?4) What happens to the orthocenter when there is a degenerate triangle (when the triangle becomes a line)?

 

Now, with GSP, create your own orthocenter of a triangle. Here are the steps below and you can use the link below to get to GSP.

GSP

1) With the mouse, highlight the point tool from the toolbox. (If you do not have a toolbox appear on your sketch go to Display on the top toolbar and go down to Show toolbox.)

2) Create three points on your sketch by clicking with the mouse.

3) Drag your mouse back to the toolbox and click on the arrow (this allows you to highlight).

4) Highlight the three points

5) Go to the top toolbar, select Construct, and go down to Segments

6) Click off the triangle to de-highlight all objects

7) Highlight one vertex and the opposite side only

8) Go to top toolbar, select Construct, and go down to Perpendicular Line

9) Click off the figure to de-highlight objects

10) Highlight another vertex and opposite side and repeat steps #8 and #9

11) Highlight the third vertex and opposite side and repeat steps #8 and #9

12) Click on the intersection of the three lines you just created. Congratulations, this is your orthocenter!

(Do not save when exiting GSP)

 

For a further exploration of the orthocenter, click below and watch what path the orthocenter takes as you drag one vertex along a line.

Explore

?5) What shape does the path of the orthocenter follow?


Circumcenter

The CIRCUMCENTER (C) of a triangle is the point in the plane equidistant from the three vertices of the triangle. Since a point equidistant from two points lies on the perpendicular bisector of the segment determined by the two points, C is on the perpendicular bisector of each side of the triangle.

We can explore the circumcenter by using Geometer's Sketch Pad. Click on the link below to explore and make observations about the circumcenter.

Circumcenter

?6) Were there any cases where the circumcenter was outside the triangle? If so, when did it happen?

?7) What happens to the circumcenter when there is a degenerate triangle (when the triangle becomes a line)?

 

Now, with GSP, create your own circumcenter of a triangle. Here are the steps below and you can use the link below to get to GSP.

GSP

1) With the mouse, highlight the point tool from the toolbox. (If you do not have a toolbox appear on your sketch go to Display on the top toolbar and go down to Show toolbox.)

2) Create three points on your sketch by clicking with the mouse.

3) Drag your mouse back to the toolbox and click on the arrow (this allows you to highlight).

4) Highlight the three points

5) Go to the top toolbar, select Construct, and go down to Segments

6) Click off the triangle to de-highlight all objects

7) Highlight all three sides by clicking on them one at a time

8) Go to the top toolbar, select Construct, and go down to Midpoints

9) Repeat step #6, then highlight one midpoint and the side it is on

10) Go to the top toolbar, select Construct, and Perpendicular Line

11) Repeat step #6, then highlight another midpoint and the side it is on, and repeat step #10

12) Repeat step #6, then highlight the last midpoint and side, and repeat step #10

13) Click on the intersection of the lines you just created. Congratulations, you have just created a circumcenter!

(Do not save when exiting GSP)

 

An extension of the circumcenter is the circumcircle. Since the vertices are equidistant from the circumcenter, we can create a circumcircle using the circumcenter as our center and the vertices as points on the circumcircle. This causes the circumcenter to the each vertex to be a radius of the circumcircle. Click below to explore.

Circumcircle


Incenter

The INCENTER (I) of a triangle is the point on the interior of the triangle that is equidistant from the three sides. Since a point interior to an angle that is equidistant from the two sides of the angle lies on the angle bisector, then I must be on the angle bisector of each angle of the triangle.

We can explore the incenter by using Geometer's Sketch Pad. Click on the link below to explore and make observations about the incenter.

Incenter

?8) Were there any cases where the incenter was outside the triangle? If so, when did it happen?

?9) What happens to the incenter when there is a degenerate triangle (when the triangle becomes a line)?

 

Now, with GSP, create your own incenter of a triangle. Here are the steps below and you can use the link below to get to GSP.

GSP

1) With the mouse, highlight the point tool from the toolbox. (If you do not have a toolbox appear on your sketch go to Display on the top toolbar and go down to Show toolbox.)

2) Create three points on your sketch by clicking with the mouse.

3) Drag your mouse back to the toolbox and click on the arrow (this allows you to highlight).

4) Highlight the three points

5) Go to the top toolbar, select Construct, and go down to Segments

6) Click off the triangle to de-highlight all objects

7) Now, you're going to select an angle. To do this, you must highlight the three vertices of the triangle with the middle one being the vertex of the angle you will bisect.

8) Go to the top toolbar, select Construct, and go down to Angle Bisector

9) Repeat step #6, select the three vertices again with the middle point being a different vertex, and repeat step #8

10) Repeat step #6, select the three vertices one last time with the middle point being the last vertex to bisect, and repeat step #8

11) Click on the point of intersection for these angle bisectors. Congratulations, you just created your own incenter!

(Do not save when exiting GSP)

 

Since the incenter is equidistant from the three sides, the incenter also has an extension called an incircle. Click below to see.

Incircle

?10) Do you have any ideas on how to create the incircle given the incenter? Solution


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