by Sharren M. Thomas
Altitudes and Orthocenters
I will begin with an exploration of orthocenters that have been constructed inside of a particular triangle, ABC.
When a triangle HBC is constructed from a triangle ABC and its orthocenter H. Where will the orthocenter be located for the new triangle HBC? Let us explore.
The orthocenter for HAB is at C. Similarly the Orthocenter for triangle HBC is located at point A. And the orthocenter for HAC is located at point B. See below the orthocenter for HAB, HBC, and HAC.
Now I will begin by constructing the circumcenters of triangles ABC, HBC, HAB, and HAC which are respectively points E, F, G, and H, see below. I will construct the CircumCIRCLES for each triangle, also see below.
Now, let's explore the relationship between the type triangle that ABC is and its relationship with the previously constructed orthocenters and circumcircles.
See below that when triangle ABC is a right triangle, where angle A is the right angle then notice that the triangle ABC's orthocenter H, corresponds with the vertex A; which happens to be the circumcenter for triangle HBC. Thus, triangle ABC and triangle HBC have the corresponding circumcenters, therefore the circumcircles are the same. If in turn, ABC is a right triangle with the right angle at vertex B, then circumcenter of ABC, point H and the circumcenter for triangle HAC point B correspond. n Similarly, ABC is a right triangle the right angle at vertex C, then circumcenter, H, or ABC and the circumcenter of HAB, C, will overlap.
If triangle ABC is an equilateral triangle, and simultaneously both its orthocenter point H and its Circumcenter, point E overlap; then the circumcenters of triangle HBC, HAB, and HAC are points located on triangle ABC's Circumcircle. See below.
Now let's construct the 9-point circle from this construction. See below, the blue circle is the 9-point circle.
For a triangle ABC. The 9-points include the 3 midpoints of the original triangle. A medial triangle is created from these midpoints. Now, if we construct the orthocenter of this medial triangle this becomes the center of the 9-point circle, whose radius is the distance from the orthocenter of the medial triangle to any midpoint of the original triangle ABC. The 9-point circle also passes through the perpendicular segments, and the feet of a triangles perpendicular.