#### Relationship between the orthic triangle

#### and the triangle created by the intersection
of altitudes and circumcircle

#### by

#### Rachael Brown

First off, I constructed the picture of the
orthic triangle and the triangle created by the intersection of
the altitudes and circumcircle in GSP.

The blue triangle is the original triangle,
the green triangle is the orthic triangle, and the red triangle
is the triangle created by the intersection of the altitudes and
the circumcircle. Through my explorations I realized that the
line through F and the orthocenter is an angle bisector. This
is true for the other lines through G and the orthocenter and
H and the orthocenter. This means that the orthocenter for triangle
ABC is the incenter for triangle FGH.

As is clear from the picture, the green and
red triangles are similar. As I measured in sketchpad, I saw that
the scale factor for the two triangles is 2. Click here
to open a GSP file to explore this.

Now, I need to prove my claim that the triangles
are similar and the scale factor is 2. From previous experiences,
I know that the orthic triangle is on the nine point circle of
triangle ABC. The nine point circle not only crosses through the
feet of the altitudes but it crosses through the midpoints of
the sides of triangle ABC.

In the above picture, the purple circle is
the nine point circle. This circle is similar to the circumcircle.
Its scale factor is 2. The radius of the nine point circle is
1/2 the radius of the circumcircle.

Because the orthic triangle is inscribed in
the nine point circle and the red triangle is inscribed in the
circumcircle they have a ratio of 1 to 2. This is not only because
of the relationship between the nine point circle and the circumcircle
but also due to the fact that vertices of the triangles lie on
the altitudes.

#### Educational Value

This experience was worthwhile in that it helped
me really think through all that I thought I understood about
the orthocenter, circumcircle, perpendicular bisectors, nine point
circle, and altitudes. I would love for my students to have the
same experience so that they have the opportunity to challenge
what they know and understand.

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