An Investigation of Triangle Centers

This page is designed so that high school students can explore triangles and their centers using Geometer's Sketchpad. They will investigate centroids, orthocenters, circumcenters, and incenters. Students may draw conclusions based on the constructions -- some of which will be theorems that are to be proven in class.

Centroid

The centroid of a triangle involves constructing all of the medians on a triangle. Let's start with an acute triangle and construct the medians. Do you notice anything special? What would happen if we started with an obtuse triangle? What about a right triangle? [Instead, of starting over with an obtuse triangle, choose a vertex of the acute triangle and drag it until it makes the triangle obtuse. Do the same for the right triangle.]

Construct any triangle and its centroid. You may want to hide the segments that are the medians. Now, construct segments from each vertex and midpoint to the centroid -- it should look exactly the same as before you hid the medians. Using the measure segment function in Geometer's Sketchpad, measure each of the segments. Do you see any relationship in the distance from the vertex to the centroid to the distance from the centroid to the midpoint? Make a hypothesis and try it out on several different triangles.

The centroid, G, of a triangle is the common intersection of the three medians.

The medians of a triangle intersect in a point that is 2/3 of the distance from each vertex to the midpoint of the opposite side.

The centroid is also called the center of gravity. If you were to cut out a triangle out of cardboard and construct its centroid, it would be perfectly balanced at this point.

Orthocenter

The orthocenter of a triangle involves constructing the altitudes of the triangle. Construct an acute triangle and its altitudes (altitudes are lines that are perpendicular to a side of the triangle and go through the opposite vertex). Do you notice anything special about the intersection of the altitudes? What would happen if we had started with an obtuse triangle or a right triangle? Make a hypothesis about each type of triangle and its orthocenter and they try it out for several triangles.

The orthocenter, H, of a triangle is the common intersection of the three lines containing the altitudes.

If the triangle is acute, the orthocenter will lie inside the triangle. If the triangle is obtuse, the orthocenter will lie outside the triangle. If the triangle is right, the orthocenter will lie on the vertex that corresponds with the right angle.

Circumcenter

The circumcenter of a triangle involves constructing the perpendicular bisectors of a triangle. Again, we will start with an acute triangle and construct the perpendicular bisector of each side (construct the midpoint of each side and then choose the side and the midpoint to construct the perpendicular line). By now, you should realize that the lines are to intersect at the same point. Do you notice anything special about the intersection of the perpendicular bisectors? Using this intersection and a vertex on the triangle, construct a circle. What do you notice now? Does changing the original triangle to an obtuse or a right change anything?

Construct any triangle and its circumcenter. You may hide the perpendicular bisectors needed to find the circumcenter. Now, construct the segments from each vertex to the circumcenter. Using the measure segment function in Geometer's Sketchpad, measure each segment. Do you notice any special relationship? Make a hypothesis and test it on several different triangles. Does it work for acute triangles? obtuse triangles? right triangles?

Let's expand on this in the case of the right triangle. Construct the circumcenter on a right triangle. Construct a segment from each vertex of the hypotenuse to the circumcenter. What relationship do you notice about the circumcenter and the hypotenuse? Make a hypothesis and test it for several right triangles.

The circumcenter, C, of a triangle is the point in the plane equidistant from the three vertices of the triangle.

The circumcenter will always coincide with the midpoint of the hypotenuse of a right triangle.

Incenter

The incenter of a triangle involves constructing the angle bisectors. We will start by constructing an acute triangle with its angle bisectors. Do you notice anything special about the intersection of the angle bisectors? What if we had started with an obtuse triangle or a right triangle?

Now, hide the angle bisectors. Construct a perpendicular segment from the incenter to any side. Using the incenter as the center and the point of intersection of the perpendicular line to the side as another point, make a circle. What do you notice about the circle? Make a hypothesis and test it for several triangles.

The incenter, I, of a triangle is the point on the interior of the triangle that is equidistant from the three sides.

Euler Line

Now, let's see if there are any relationships between these centers of a triangle. Construct any triangle with the centroid, orthocenter, circumcenter, and incenter labeling them G, H, C, and I, respectively. Construct a segment between each pair of points. Your construction may be too cluttered, so you may delete the incenter for now - we will return to it later. Do you notice a special relationship in these points?

The centroid, circumcenter, and orthocenter always lie on a straight line -- the Euler Line.

Do you notice any special relationship between the distances between the points?

The distance from the orthocenter to the centroid is always twice as long as the distance from the centroid to the circumcenter.

Now, add the incenter to the construction. Do you notice anything special about the incenter with its relation to the Euler Line?

The incenter lies on the Euler Line if the triangle is isosceles.

What happens to the points if the triangle is equilateral?

If the triangle is equilateral, the points are concurrent.

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