Altitudes and Orthocenters
We want to investigate the relationships between altitudes and orthocenters.
Our exploration centers around the question:
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.
Next construct the orthocenter, H, of triangle ABC. Recall the orthocenter
of a triangle is the common intersection of the three lines containing the
For a GSP script that constructs the orthocenter of any triangle, click
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.
The orthocenter of triangle HAB is C.
The orthocenter of triangle HAC is B.
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 construct the orthocenter of HAC call it B.
When the orthocenter B is constructed a new triangle ABC is also constructed
in the process where B is a vertex of this new triangle ABC. Furthermore,
two more new triangles are formed from this construction, triangles HAB
and HBC. If we look at triangle HAB and HBC individually and construct their
orthocenters, we would find their orthocenters to be vertexes of the triangle
ABC. Also the point H, the common vertex of the three triangles HAC, HAB,
and HBC, is the orthocenter of triangle ABC.
Next consider the construction of the circumcircles of triangles ABC,
HBC, HAB, and HAC.
Click here for an animation as triangle ABC
varies by shape and size.
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
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.
Notice points A, B, and C are all on one segment and the altitudes have
been transformed to perpendicular lines to segment AC through the points
A, B, and C.
Revisit our picture with the circumcircles of triangles ABC, HBC, HAB,
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,
What do you discover?
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.