Ornery Orthocenters

Assignment 8

By Amber Candela

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What is an orthocenter?

The orthocenter of a triangle is the intersection of the altitudes of the triangle. An altitude of a triangle is the segment that is perpendicular from a vertex to the opposite side. If the triangle is acute, the orthocenter will remain on the inside. If the triangle is obtuse the orhocenter will be on the outside. If the triangle is a right triangle the orthocenter will be on the vertex of the right angle.

Let's construct a triangle and investigate the orthocenters.

Construct triangle ABC with orthocenter H. Then construct the orthocenter of triangles ABH, ACH and BCH, using two of the vertices and the orthocenter H. Where is the orthocenter for each triangle?

The orthocenter for each triangle with H as a vertex is the vertex from triangle ABC that is not included in the triangle. For example, the vertex for triangle ABH is point C.

Why does this occur?

Take triangle ABH. Vertex B needs to be perpendicular AH. AH has already been constructed to be perpendicular to BC, and there can only be one perpendicular to a given line, so B will be perpendicular to AH on segment BC. Vertex A needs to be perpendicular to BH. BH has already been constructed to be perpendicular to AC so A will be perpendicular to BH on segment AC. Vertex H needs to be perpendicular to AB. HC has already constructed to be perpendicular to AB so H will be perpendicular to AB on segment HC. The vertices will be perpendicular to the opposite sides on the following segments, AC, BC and HC, C is the common point to all of those segments thus making the orthocenter C. This follows similarly for each other triangle.

Let's make a switch!

If you switch the orhocenter with one of the vertices what happens? The vertex becomes the orthocenter and the orthocenter will become the vertex.

Let's Make Some Circumcircles!

What is a circumcircle?

A circumcircle is the circle that circumscribes the triangle. This means each of the triangles vertices are on the circle. If we create a circumcircle for each of the four triangles we see the following picture.

How is this all connected?

When you connect the circumcircle centers of the three smaller triangles, it creates triangle LMN. When you find the orthocenter of triangle LMN it is P which is the center of the circumcircle of triangle ABC. This makes the circumcenter of the original triangle, the orthocenter of the triangle created by the centers of circumcircles of smaller triangles.

Conjectures:

The following are conjectures that can be made from the construction:

• The four circumcircles created are all congruent.
• The area of the original triangle ABC is the same as the area of triangle LMN.

One Cool Thing.....

If you connect the vertices of both triangles, a hexagon is created!

Can you move the vertices around until you create a regular hexagon? When will this occur?