Problem 5: Use GSP to construct G,H,C,and I for the same triangle. What relationships can you find among G,H,C,and I or subsets of them? Explore many shapes of triangles.
G is the center of gravity of a triangle formed by the intersection of the medians. To see the construction of G, click here.
H is the orthocenter of a triangle formed by the intersection of the altitudes. To see the construction of H, click here.
C is the circumcenter of a triangle formed by the intersection of the perpendicular bisectors. To see the construction of C, click here.
I is the incenter of a triangle formed by the intersection of the angle bisectors. To see the construction of I, click here.
The Euler Line is line that passes through points G,H and C. To see the four centers in relation to the Euler Line, click here.
Feel free to investigate for yourself the relationships among the four center points. You may do this by clicking here and then by moving around the vertices of the given triangle.
Below I have listed some of my observations.
Centers G,H, and C are always on the Euler Line.
When the triangle is an obtuse triangle, centers H and C are outside the triangle. These two points move closer to the sides of the triangle as the triangle becomes closer to an acute triangle.
These two points both touch the triangle at the same time. Can you guess when? When it becomes a right triangle. The orthocenter is on the vertex of the right angle and the circumcenter becomes the midpoint of the hypotenuse.They move inside the triangle when the triangle becomes an acute triangle.
The center I is also on the Euler Line if the triangle is an isosceles triangle. This obviously is also true if the triangle is an equilateral triangle because at least two sides are equal.
Let's look at an equilateral triangle. What do you notice that happens? All of the points converge to the same point, which just happens to be in the dead center of the triangle.
What happens to the incenter when the triangle becomes degenerate? It becomes the same point as the vertex that is between the other two vertices.
The distance from G to C is one third the distance from G to H no matter what happens to the triangle.
If you don't believe me on these observations, check them out for yourself.