Assignment 4:
Triangle Centers
by:
Sarah Link
Where is the center of a triangle? The answer to this all depends on what kind of center you are looking for. There are four centers of every triangle: the incenter, the orthocenter, the centroid, and the circumcenter. They all are found different ways and all serve different purposes. Let's take a look first at what each center is and how we can construct each of them.
The Centroid:
The centroid is one that can easily be applicable to the real world - it is our center of mass of the triangle. If you were to have a cardboard triangle and wanted to balance it on a pinpoint, the centroid is where you would need to place your pin point. We find the centroid by intersecting our three medians of the triangle. As a reminder to earlier lessons on triangles, the median is the line connecting a vertex to the midpoint of the opposite side. Every triangle has three medians and there are concurrent at one point labeled as the centroid, commonly denoted as point G. Below we have pictures of a triangle first with just the midpoints highlighted and then with midpoints connected with their opposite vertices. The centroid is labeled as G.
The animation included in the GSP link below will display how the centroid moves as the shape of the triangle changes by animating point A:
We see on this animation that the centroid, labeled G, remains within the triangle ABC no matter the shape the traingle takes.
The Orthocenter:
The orthocenter is the point of concurrency between the three altitudes of a triangle. An altitude is the line that passes through a vertex and is perpendicular to the side opposite that vertex. Here we can see where the orthocenter lies in triangle ABC. The orthocenter is denoted by H.
Now we want to look at how the orthocenter changes when the shape of the triangle changes. Again we have a GSP animation to help illustrate what happens to H as the triangle changes shapes by adding motion to point C:
From this animation we see that the orthocenter travels inside, outside, and even along the triangle it is constructed from. When the triangle is right, it lies on a vertex, when it is acute (as in the picture above) it remains within the walls of the triangle legs. However, when the triangle becomes obtuse, the orthocenter travels to the outside of the triangle yet still occurs in only one place of concurrency.
The Circumcenter:
The circumcenter, labeled C, is the center of the circumscribed circle to triangle ABD. We construct this center by drawing the three perpedicular bisectors and finding their intersection. The perpendicular bisector is found by drawing lines through the midpoint of a leg and perpendicular to the leg the midpoint lies on. Below the circumcenter is shown as well as the circumscribed circle. It can be seen that this circle goes through all of the vertices of the triangle. This being so, if we wanted to we could draw radii from the point C to each of the vertices and by the rules of a circle, these radii would all be of equal length. Therefore we can conclude that the circumcenter is equidistant from all three vertices.
In the acute triangle above, the circumcenter lies inside the triangle. Animate the point A on the GSP construction below to see how it falls on the vertex of a right triangle and outside the triangle when the shape becomes obtuse:
The Incenter:
The last triangle center we want to look at is the incenter, or I. To construct the incenter we take the intersection of the three angle bisectors. This gives us the center point of our inscribed circle. Just as we saw above that the circumcenter is equidistant from the three vertices, the incenter is equidistant from each of the three sides. The picture below shows the triangle ABC, the incenter I, and the inscribed circle drawn as well:
Now we want to animate point C to see that as you may conject, the incenter always remains within the walls of the triangle:
Now let's take a look at all four of these centers on one triangle and watch as the triangle moves to see the relation between the four by animating point A.
By taking a few snap shots from the animation we can take a closer look at what is going on between the four centers and when special relationships are occurring. In an acute triangle we see that all four lie within the triangle and that H and C and G all lie on one line segment that can be drawn from H to C. This segment is known as the Euler Line and we will examine which centers fall along it in each triangle shape.
Now we will try an obtuse triangle. Here H and C fall outside the triangle, but the segment linking them still contains our centroid G.
In the isosceles triangles below, all four centers fall on our line and all four are also contained within the walls of the triangle.
Below we see a right triangle which still has all four centers inside, but the incenter has fallen off the Euler line.
Finally, we have an equilateral triangle in which not only do all four points fall along our line and within the triangle, they are in fact all four concurrent at one point in the center of the triangle.
Now that we've examined where these centers fall in the different types of triangles, we want to explore even further to something known as the Nine-Point Circle. The Nine-Point Circle for any triangle passes through the three mid-points of the sides, the three feet of the altitudes, and the three mid-points of the segments from the respective vertices to orthocenter. Below we have these nine point labeled on a given triangle ABC. The mid-points of the sides are labeled with M's, the feet of the altitude with A's, and the mid-points of the segments from the vertices to the orthocenter with O's. All nine points are highlighted.
We now see the nine points that help define this circle and contribute to a lot of its interesting qualities, but we are not able to draw it solely from the information above. These points all lie on the circle, but are we still do not know where the circle's center lies.