The line created by the orthocenter (H), centroid (G), and the circumcenter (C) of a triangle is reffered to as Euler's Line. The endpoints of this line are always H and C, with G found between these two points. This line also contains relationships to a point called the incenter (I). To investigate the relationships present in the Euler Line and the incenter, we need to examine various different types of triangles.
First we will use Geomter's Sketchpad to create these four points for a random triangle which can be manipulated into different types of triangles. Then we will connect the point H, G, and C to form a line segment. These actions will create the picture shown below. The class will notice that H, G, and C indeed form a straight line. Try moving the triangle around and see if this appears to be true for all triangles.
What do you think the relationship between HG and HC is?
Click here to see the answer
Does anything unusual happen when the triangle is a right triangle? Construct a right triangle and observe the relationships.
For all right triangles, the centroid is located on a midpoint of the hypotenuse and the orthocenter is located on the vertex across from the hypotenuse. Why do you think this occurs?
Let's investigate the difference between an acute and an obtuse triangle.
Did you notice that the Ueler Line is contained inside the acute triangles and outside the obtuse triangles? Which points remain inside the triangle no matter what and which points can be both inside and outside the triangle? When and where do those points exit the triangle? To see the answers, click here.
We will now explore the relationships in an isosceles triangle. The picture below shows that when the triangle is isosceles, the Euler's Line is located along the perpendicular bisector of the non-equal side. Itis also interesting to note that the incenter falls on the Euler's Line in this triangle, while it did not in any of the other triangles we explored. Why do you think that is?
Another interesting relationship to be explored is the relationship of this line when the triangle is an equilateral triangle. Manipulate the sides of the triangle and examine what happens as the lengths of the side approach equality. As the lengths of the sides of triangle ABC approach equality, H, G, and C converge to one point, which converges also to the incenter I.