This write-up is based
on an exploration of any triangle (**RST**) and its medial triangle (**DEF**). A medial triangle is one that
is formed by connecting the midpoints of the sides of the original
triangle. It is similar to the original triangle and one-fourth
of its area.

Specifically, the exploration
focuses on how the centroid (**G**),
orthocenter (**H**), circumcenter (**C**), and incenter (**I**) of the original triangle and
the medial triangle compares for different shaped triangles.

For each case shown below,
**G**, **H**,
**C**, and **I**
were used to represent the special points for the large triangle
**RST**. Points **G', H'. I', **and**
C'** were used for the medial triangle **DEF**.

As I began the exploration
and constructed the centroid (**G**),
I noticed this point was concurrent fro both triangles and remained
that way with different shaped triangles. Therefore, only one
label (**G**) was used for both triangles.
this can be seen in all the cases shown below.

The construction of the
orthocenter of the medial triangle shows the point concurrent
with that of the circumcenter for triangle RST
and remains when exploring different shaped triangles. Notice
that **C** and **H'** are concurrent
for all the cases shown below.

**click here
to investigate GSP sketch**

**G**)
concurrent

**H'**) of triangle
DEF is concurrent with circumcenter (**C**)
of triangle **RST**

**I**
and **I'**) are not concurrent

**click here
to investigate GSP sketch**

**G**)
is concurrent

**H'**) and circumcenter
(**C**) remain concurrent

**click here
to investigate GSP sketch**

**click here
to investigate GSP sketch**

**click here
to investigate GSP sketch**

**click here
to investigate GSP sketch**

**Summary**

**RST** and the orthocenter of triangle
**DEF** remain concurrent for all cases.

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