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Altitudes and Orthocenters

by

Tonya DeGeorge

 

To begin this investigation, letÕs construct a triangle ABC, as shown below:

And then construct the orthocenter, H (recall that the orthocenter is the intersection of the perpendicular bisectors of each vertex to the opposite side):

Therefore, triangle ABC can be broken up into three smaller triangles: HBC (purple), HAB (yellow), and HAC (light blue), as shown below:

Now suppose we find the orthocenter of each new triangle that was formed by the orthocenter of the original triangle ABC.  What do you think we will find?  LetÕs begin by finding the orthocenter of triangle HBC.

We can see that the orthocenter of triangle HBC is in fact the vertex A of the original triangle.  Do we get similar results when we find the orthocenter of triangles HAB and HAC?  Below are the sketches that show the orthocenter of both triangles:

 

We can see that the orthocenter of each triangle is one of the vertices of the original triangle.  Our findings are shown below:

Now suppose we construct the circumcircles of triangles ABC, HBC, HAB, and HAC.  What do you suppose we will find?  (Recall that the circumcenter is formed by the intersection of the perpendicular bisectors of the triangle.  By finding the circumcenter first, it will be easier to construct the circumcircle.)

If we move the vertex A towards the orthocenter H, we can see that the two circumcircles overlap and become one circle, as shown below:

Likewise, we get the same thing when we move the vertices B and C on top of the orthocenter.  What type of triangles form?

Every time we move one of the vertices onto the orthocenter, H, we get a right triangle.  We know this because the inscribed angle, H in each case, encompasses the 180 degree arc (and we know that the inscribed angle is ½ the arc measure).

Now suppose we connect the vertices of the triangle and the circumcenters of the interior triangles, we get a picture like this:

 

The shape of this six-sided figure (hexagon) changes as we move the vertices of the original triangle.  The hexagon becomes a regular hexagon when we move the orthocenter, H, on top of the circumcenter of the original triangle ABC:

In conclusion, we have found:

¯ The orthocenters of triangles (HBC, HAB, HAC – which were formed by the orthocenter of the original triangle ABC) are the vertices of the original triangle ABC

o  Vertex A is the orthocenter of triangle HBC

o  Vertex B is the orthocenter of triangle HAB

o  Vertex C is the orthocenter of triangle HAC

¯ When we move one of the vertices of the original triangle ABC to the orthocenter, H, right triangles are formed.

¯ When you connect the vertices of the original triangle with the circumcenters of the interior triangles, it will form a six-sided figure

o  This figure becomes a rectangle when we move the vertices of the original triangle to the orthocenter.  The right triangle formed is half the area of the rectangle.

o  This figure becomes a regular hexagon when the orthocenter becomes the circumcenter of the original triangle ABC.  Triangle ABC is half the area of the regular hexagon.

 

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