Can we find interesting mathematical phenomena and make connections among the altitudes and orthocenters of a given set of triangles?

Our exploration begins with the following construction:

First, construct any triangle ABC.

Now consider the triangle HBC. Construct the orthocenter of triangle HBC.

Did you notice anything interesting?

When constructing the orthocenter for HBC, the altitude from H to BC is the same as the altitude from A to BC. Also, the altitude from B to HC lies on segment AB, one of the sides of the original triangle ABC. Similarly, the altitude drawn from C to HB is also segment AC. The most interesting of all is the orthocenter of triangle HBC is A, one of the vertexes of the original triangle ABC.

Does this same pattern repeat itself when constructing the orthocenters for triangles HAB and HAC?

If our conjectures are true, the orthocenter of triangle HAB should be C and the orthocenter of triangle HAC should be B.

Construct the orthocenters of triangles HAB and HAC to verify.

Let's also look at what happens when we reverse the order of constructing the original triangle ABC when the orthocenters for each individual triangle are constructed from the orthocenter of triangle ABC.

For example, start with triangle HAC.

Next consider the construction of the circumcircles of triangles ABC, HBC, HAB, and HAC.

As triangle ABC is changing, notice what happens when triangle ABC becomes a right triangle. H becomes B and their circumcircles overlap. Remember in a right triangle, the vertex at the right angle is also the orthocenter. So it makes sense that their circumcircles are the same.

If we continue to push point B further in towards segment AB to make an obtuse triangle, the orthocenter shifts from inside triangle ABC to outside of the triangle where the altitudes meet. Remember in an obtuse triangle, the orthocenter always lies outside of the triangle. Why? The intersection of the extended altitudes intersect outside of the triangle.

What will happen if we continue pushing B further in until it lies on segment AC?

Click here for an animation as B moves closer to segment AC.

When point B moves so close that it lies on segment AC and triangle ABC no longer exists, the following occurs.

Revisit our picture with the circumcircles of triangles ABC, HBC, HAB, and HAC.

Recall the Nine-Point Circle of any triangle passes through the three mid-points of the sides, the three feet of the altitudes, and the three mid-points of the segments from the respective vertices to the orthocenter. The center of the Nine-Point Circle is the midpoint of the segment whose endpoints are the orthocenter and the circumcenter.

For fun, construct the Nine Point Circles for each triangle ABC, HAB, HBC, and HAC.

Most fascinating, the Nine-Point Circles for all four triangles are the SAME. The center for the Nine-Point Circles is the midpoint of the segment whose endpoints are the orthocenter and the circumcenter. The center of the Nine-Point Circle also lies on the Euler Line. The Euler line is the line containing the centroid, circumcenter, and orthocenter of a triangle.

As triangle ABC moves and changes shape and size the Nine-Point Circles still remain the same for triangles ABC, HAB, HBC, and HAC.

Click here for an animation.

Constructing the Nine-Point Circle for the triangles ABC, HAB, HBC, and HAC is a nice extension that ties together the concepts of altitude, orthocenter, centroid, Nine-Point Circle, and the Euler Line. The construction stresses the interrelatedness of all these geometrical concepts.

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