Classical Triangle Centers

The sketches below illustrate the four classical triangle centers. In each sketch, you should drag around the vertices of the triangle, observe the behavior of the triangle center, and answer the questions next to each sketch.

The Orthocenter

The orthocenter is the point of concurrency of the three altitudes.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) The sketch at left illutrates the orthocenter. As you drag around the vertices, answer the following questions:

1. Is it possible to have the orthocenter OUTSIDE the triangle?

2. Where is the orthocenter located in an ACUTE triangle?

3. Where is the orthocenter located in an OBTUSE triangle?

4. Where is the orthocenter located in a RIGHT triangle?

5. If you made a triangle using A, B, and the orthocenter, what would be the orthocenter of THAT triangle?

The Incenter

The incenter is the point of concurrency of the three angle bisectors.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) The sketch at left illutrates the incenter. As you drag around the vertices, answer the following questions:

1. Is it possible to have the incenter OUTSIDE the triangle?

2. Where is the incenter located in an ACUTE triangle?

3. Where is the incenter located in an OBTUSE triangle?

4. Where is the incenter located in a RIGHT triangle?

5. What does the circle tell you about the incenter's relationship to the sides of the triangle?

The Centroid

The centroid is the point of concurrency of the three medians.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) The sketch at left illutrates the centroid. As you drag around the vertices, answer the following questions:

1. Is it possible to have the centroid OUTSIDE the triangle?

2. Where is the centroid located in an ACUTE triangle?

3. Where is the centroid located in an OBTUSE triangle?

4. Where is the centroid located in a RIGHT triangle?

5. Can the centroid ever be closer to the vertex of a triangle than it is to the opposite side?

The Circumcenter

The circumcenter is the point of concurrency of the perpendicular bisectors.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) The sketch at left illutrates the circumcenter. As you drag around the vertices, answer the following questions:

1. Is it possible to have the circumcenter OUTSIDE the triangle?

2. Where is the circumcenter located in an ACUTE triangle?

3. Where is the circumcenter located in an OBTUSE triangle?

4. Where is the circumcenter located in a RIGHT triangle?

5. What does the circle in the sketch tell you about the circumcenter's relationship to the triangle's vertices?

Steve Phelps, Created with GeoGebra