Constructing the Euler Line in Geogebra

In this post, we are going to see how to construct the Euler line in Geogebra - an online graphing calculator. But first, a short description of what the Euler line is. 

In any triangle, the four notable centers - centroid, orthocenter, circumcenter and nine-point center - are collinear; meaning a straight line can connect all four of them. This line is the Euler line of that triangle. 

To see that by ourselves, we are going to use 2D graphing calculator in Geogebra to construct these four centers. So here’s a step by step guide on how to draw the Euler line passing through these four centers of a triangle in Geogebra (demonstrated with pictures). 

We are gonna need a triangle to begin with. I will assume you already know how to construct triangles in Geogebra. But if you want to know how to construct triangles of three given side lengths or two given side lengths & one given angle in Geogebra, check out this post of mine. 

Step 1 : Drawing a triangle. Select the tool Polygon under the Polygons section in the Shapes menu. 

To draw a triangle, click on any three different spots in the graphing panel to the right to mark the locations of the three vertices of the triangle. Click again on the first point you made to finish forming the triangle. This is probably the easiest way to draw any casual triangle.

(I have kept the axes and the grid hidden in the graph for the clarity of picture) 

Step 2 : Constructing the nine-point center. But first we are going to need the nine-point circle whose center is the nine-point center. The nine-point circle of a triangle is a circle passing through the following nine points :

 

• Feet of the three altitudes

• Midpoints of the three sides

• Midpoints of line segments from the vertices to the orthocenter of the triangle. 

But to draw the nine-point circle, we don’t require all of the nine points. Any three of them will do because only three points(non-collinear) are needed to construct a circle. Since midpoints are the easiest to find we will use them to draw this circle. 

Select the tool Midpoint or Center under the Construct section. Once this tool is selected, find the midpoints of the three sides of your triangle. To find the midpoint of any side, say AB, just click on the vertices A and B in any order. 

In our example D, E & F are the midpoints. 

Next up, select the tool Circle through 3 points in the Circles section. Click on the three midpoints of the sides in any order to construct the nine-point circle though them. 

The circle will not have its center marked. We need to locate where its center is. There are multiple ways in which we could do that. One way is by using the tool named Extremum in the Basic tools section. Select this tool and click on the nine-point circle.

What this tool will do is mark a pair of points on a curve that are farthest away from each other horizontally(G & H). It will also mark a pair of points on a curve that are farthest away from each other vertically(J & I). But since the curve is a circle, if we join the two points in a pair by a segment, that segment will be the diameter of the circle. And the two diameters intersect at the center of the circle. 

So the next thing that we will do is use the tool named Segment under the Lines section. Click on the points G & H in any order to join them with a segment. Similarly join the points J & I with a segment. 

Notice that in my case the names of these points have changed to G1, H1, J1 and I1. Frankly, i don’t know why it has happened with me and i don’t know if the same will happen to you as well. But regardless we have got the location of the nine-point center which we are looking for. To mark this point, select the tool Intersect under the Basic tools. Click on the two diameters and there will appear the point(dot) representing the nine-point center. Point K is the nine-point center in the figure below.

We are only in the second step and the graph is starting to look a little messy. To draw the Euler line, we only need the nine-point center, not the circle or the extreme points or the diameters. We can hide all the unwanted stuff. In the editing panel, click on the calculator shaped icon on top and hide the stuff that you don’t want to be visible. 

Step 3 : Drawing the Circumcenter. Circumcenter is the center of the Circumcircle - a circle that passes through the three vertices of a triangle. Select the tool Circle through 3 Points again and click on the already marked three vertices of the triangle to draw the circumcircle. 

We can locate the position of its center in exactly the same way we located the position of the center of the nine-point circle. Also, hide the unwanted stuff cluttering the graph. Point P is the circumcenter in the figure below. 

Step 4 : Constructing the orthocenter. Orthocenter is the point where all the three altitudes of a triangle intersect. To draw an altitude, select the tool Perpendicular Line in the Construct section. Click on a vertex and the side opposite to it. 

Click on any other vertex and the side opposite to it to draw a second altitude which will intersect the first altitude at a point. 

The third altitude will also pass through the same point but we don’t need to draw it as we already have found out where they are intersecting. Mark this point using the Intersect tool in the basic tools section and clicking on the two altitudes. You can hide the altitudes after you have marked the orthocenter. 

Step 5 : Constructing the Centroid. Centroid is the point where the three medians of a triangle intersect. I did not find a tool that will help us directly draw a median like the Perpendicular Line tool for an altitude. But we can use the Segment tool to draw a median. 

A median is a line segment joining a vertex to the midpoint on the opposite side. So first we can use the Midpoint or Center tool to find/mark the midpoints on the sides. But we have already done that in the second step, we used the midpoints to draw the nine-point circle. We just have to unhide the midpoints(if you had kept them hidden). 

After they are visible again, use the Segment tool to connect a vertex with the midpoint on the opposite side. That’s a median. Similarly draw one more median to get the location of the centroid. 

Finally mark the centroid using the Intersect tool and clicking on the two medians. This is how your figure will look like(after hiding all the unwanted stuff).

Where, 

K ⇒ Nine-point Center

P  ⇒ Circumcenter

Q ⇒ Orthocenter

R ⇒ Centroid

Step 6 : Using the Line tool under the Construct section, we can draw a line through them. Just click on any two of the four centers and there will appear a line passing through all the four centers. That’s the Euler Line of your triangle.