Investigations of the Medial and Orthic Triangles

by: Lauren Wright

Before we begin this investigation, let's illustrate and define the triangles we will be working with.

TRIANGLE 1

ORIGINAL TRIANGLE

TRIANGLE 2

MEDIAL TRIANGLE

TRIANGLE 3

ORTHIC TRIANGLE

TRIANGLE 4

H is the orthocenter of TRIANGLE 1.

The vertices of TRIANGLE 4 are found by taking the midpoints of HA, HB, and HC.

Now, let's find the circumcircle of TRIANGLE 2, TRIANGLE 3, and TRIANGLE 4.

TRIANGLE 2

TRIANGLE 3

TRIANGLE 4

Now, let's take a look at all of the triangles and their circumcircle(s).

Clearly, all of the circumcircles are the SAME!

WHY IS THIS TRUE?

Recall how a circumcircle is formed. It is the circle whose center is the circumcenter of the triangle in question. The circumcenter is the intersection of all of the perpendicular bisectors of each leg of the triangle.

For example, the circumcenter, X, of TRIANGLE 3 would be found like so:

Now, let's look at the circumcenter of all the triangles.

TA-DA! This explains why all of the circumcircles are the same, they all share the same circumcenter!

It is also interesting to note that in constructing these three secondary triangles and their circumcircle, we have also constructed the nine-point circle for TRIANGLE 1.

The nine-point circle for any triangle passes through the three mid-points of the sides, the three feet of the altitudes, and three mid-points of the segments from the respective vertices to the orthocenter.

By definition, the nine-point circle is made up of the vertices of each secondary triangle that we constructed here!