__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**

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!