In this investigation, we are going to explore the orthocenters of triangle ABC and the orthocenters of the triangles created from the orthocenter (H). In the sketch below, I have constructed triangle ABC and it's orthocenter. The orthocenter is created from the intersection of the altitudes of any given triangle. (In other words, construct a perpendicular line from each vertex through the opposite side of the triangle.) So, the orthocenter H of triangle ABC is the center of the altitudes.
Now, let's see where the orthocenters will be for the 3 new triangles created by the altitudes. Where do you expect them to be? Because the three new triangles: HAB, HAC, and HBC are all obtuse, we can expect the orthocenters to be outside of the triangle. Also, think about where the orthic triangles originated from. To open a gsp sketch of the triangle above, click here.
Do you have an idea where the orthocenter of HBC will fall. Judging from the two cases above, it will probably fall on vertex A. Let's check just to be sure.
So, we can now prove our conjecture, that
the orthocenters of the orthic triangle lie on the vertices of
the original triangle, specifically on the vertex opposite.
Now, let's see if the circumcircles of a triangle have anything in common. Below is the circumcircle of a triangle ABC. The circumcircle is a circle constructed by selecting the circumcenter of a triangle (the circumcenter will be the center of the circle) and any vertex. Use that segment as the radius and the circle should connect the three vertices.
In the following sketch, I have constructed the circumcircles of the three orthic triangles. I first found the circumcenter of each (which if you scroll down you will find a brief investigation with those!) and then created the circumcircles.
To me, it looks like the circles are the same size. How can we prove this? What if we switch the vertices of triangle ABC with H(the orthocenter)? Let's give it a try. Click here for a GSP file where you can "move" these points.
WHen you move A to H and C to H and B to H, you can see that the circumcircles are the same size. So, when the orthocenter is one of the vertices of the original triangle it is apparent that the circumcircles of the triangles are congruent. This seems to hold true however, even if the orthocenter is not one of the vertices of the triangle. Therefore, the circumcircle of a triangle and its orthic triangles are always congruent.
Let's take a look at the circumcenter of triangle ABC and it's orthic triangles. First of all you need to remember that the circumcenter originates from taking the perpendiculars from the midpoints of the sides. Where the perpendiculars meet, is the circumcenter. The circumcenter is defined as the point in the plane equidistant from the three vertices of the triangle.
Look at the circumcenter of ABC.
Is it where you expected it to be? Look back to assignement 4 on my home page if you need help with constructing the circumcenter. Take a second to think about where you think the circumcenters of HAB, HAC, and HBC will be. I have already constructed the midpoints of all the segments, so that should help you. For my guess, I will say that the circumcenter of HAB will be a point to the upper - left of the triangle. I believe it will be a point on the line perpendicular to segment AB. What do you think? Look at the construction below to see if your conjecture is correct.
Is the circumcenter of HAB where you expected? Observe what you can about this point. I will write some of my observations and you can add any you think will help.
Now let's add the circumcenter of HBC to our investigation. Where do you think this will be positioned? Take an educated guess!
DId you guess correctly? I am sure you did. Well, the same things are true for the circumcenter of HBC that were true for HAB. Remember the definition of circumcenter : the point equidstant from the vertices of the triangle. Does this help you conclude why the points are located outside the triangles.
Look at the construction below. This construction shows all the circumcenters.
I have connected the three orthic triangle circumcenters and it proves very interesting! Would you have expected this from all of our investigations. I guess we had to know they were related in some way, because of the relation between the circumcircles and orthocenters. Now you can make a pretty cool star, if nothing else!
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