**6. Take any triangle. Construct
a triangle connecting the three midpoints of the sides. This is
called the MEDIAL triangle. It is similar to the original triangle
and one-fourth of its area. Construct G, H, ,C and the I for this
new triangle. Compare to G,H, C, and I in the original triangle**

Below is a triangle constructed
using Geometric Sketchpad. The triangle is represented by points
**A**,**B**, and
**C** and their corresponding line segments.
A **line segment **is a closed interval corresponding to a
finite portion of an infinite line. Line segments are generally
labeled with two letters corresponding to their endpoints, say
A and B, and then written AB. The length of the line segment is
indicated with an overbar. Line
segment **AB** is in **red**,
line segment **BC** is in **yellow**,
and line segment **CA** is in **blue**.

Next, I constructed the midpoints
of the three sides of the triangle. The **midpoint **is the
point on a line segment dividing it into two segments of equal
length. Click
here if you would like to learn more about midpoints.

See the graph below for the
for the following midpoints. The point **F **is the midpoint of **AC**, **E** is the midpoint of
**BC**, and **D** is the midpoint of **AB**.

Below, the points **D**, **E**, and **F**
are the midpoints of their corresponding line segments. Connecting
these three midpoints forms a triangle which is shown below as
the **triangle ****DEF**. This triangle is known as the **MEDIAL **triangle.
It is considered similar to the original triangle which is **triangle**
**ABC**. It is considered similar to the original triangle
because the **MEDIAL** triangle is one-fourth of the original
triangle's area. In addition, the side lengths of the **MEDIAL**
triangle are one-half the length of their corresponding similar
side of the original triangle.

Click here to take a more in-depth look at a **MEDIAL**
triangle from Math World's website. Let's do some calculations
to see if this is true!

We want to explore the two triangles
to see if, in fact, the area of the **MEDIAL** triange is one-fourth
the area of the original triangle's area. We also want to see
if the length of the similar sides of the **MEDIAL** triangle
are one-half the length of the original triangle. Let's go!

The formula for the **Area
of a triangle is A=1/2 (base)(heighth)**. Click **here**
to learn more about calculating the area of a triangle. For the
area of the original triangle **ABC**, I took the length of the base of **BC **which was (from above)
9.57 cm. Then after constructing a perpendicular line from point
**A** to the base **BC **at the point **G**, I found this distance to be 6.82 cm.

Plugging these
figures into the Area formula gives us **A = 1/2(9.57 cm)(6.82
cm)=32.6337 cm**. This is the **Area** of the original triangle
**ABC**.

Now, let's compute
the area of the MEDIAL triangle to see if it is one-fourth the
area of the original triangle. For the MEDIAL triangle, I took
the length of the base of **DF**
which was 4.78 cm. Then I constructed a perpendicular line from
the point **E** to the base of **DF **at the point **H** and found this distance to be 3.41
cm from above.

Plugging these
figures into the Area formula gives us **A = 1/2(4.78 cm)(3.41
cm) = 8.1499**. This is the **Area** of the Medial triangle
**DEF**.

So, is the Area of the Medial triangle one-fourth the area of the original triangle?

Is (1/4)(32.6337) = (8.1499)??

Yes, so the first part of our exploration proves true. The Area of the Medial triangle is one-fourth the area of the Original triangle.

Next, we want to compare the side lengths of the original and medial triangles. We want to explore to see if the sides of the medial triangle are one-half the length of the corresponding similar sides of the original triangle.

At a first glance, I thought
that the sides **CA** and **DE**,
**BA** and **EF**,
and **BC** and **DF**
were all similar sides. That is why they are color coordinated
below as such. Each pair of the similar sides appear to be parallel
with one another. After comparing the side lengths, we will be
able to determine if, in fact, the sides are similar.

It does appear
that the sides I chose were similar sides. So, we have compared
the Areas and the similar side lengths of the two triangles. Both
explorations show that the Triangle **DEF
**is a
Median triangle because it has the necessary characteristics.

For a further
investigation, let's compare the **centroid (G)**, **orthocenter
(H)**, **circumcenter (C)**, and **incenter (I)** of the
two triangles. Click **here** to learn more about
these constructions and how to sketch them.

The ** centroid
(G)** of a triangle is the common intersection of the three
medians. Below is the construction of the centroid for the Original
triangle and the Medial triangle. It is th epoint in the very
center of both triangles. As you can see, the point of the centroid
appears to be the same for both triangles. This is an interesting
observation.

The ** orthocenter
(H)** is the intersection of the three altitudes of a triangle.
The

Next, I wanted to find the Orthocenter of the Medial Triangle as well. Below is the construction of the graph with Orthocenters for both triangles.

Since it is hard to see the orthocenters in the above graph, I hid the perpendicular lines and reconstructed the graph below for clarity.

The **O-Medial** represents
the Orthocenter of the Medial Triangle and the **O-Original **represents
the Orthocenter of the Original (larger) triangle. Let's take
a closer look to see how the orthocenters of the two triangles
are related.

I decided to construct the distance from the orthocenter for each triangle to each side of the corresponding triangle. The distances are given below:

Recalling that the sides **CA and DE**, **BA and EF**,
and **BC
and DF** are similar sides,
I compared the distances for the similar sides to their related
orthocenters. It seems that the distance from **CA** to the orthocenter is **3.70 cm** which is
half of the distance from **DE **to
the orthocenter or **1.85 cm**. After further investigation,
it appears that this is the case for the other distances as well.

The ** circumcenter (C)**
of a triangle is the point in the plane equidistant from the three
vertices of the triangle. Since a point equidistant from two points
lies on the perpendicular bisector of the segment determined by
two points, the Circumcenter (C) is on the perpendicular bisector
of each side of the triangle. The circumcenter is also the center
of the circumcircle or the circumsctribed circle.

Below is the circumcenter of
the original triangle **ABC**. It was was constructed
by forming a perpendicular line at each of the line segment's
midpoint. The perpendicular lines are shown in green below.

Next, we want to construct the circumcenter of the Medial triangle to compare to the circumcenter of the Original triangle. The lines perpendicular to the midpoint of the Medial triangle are shown below.

As you can see from above, the distances for the related triangles from each point to each circumcenter is the same. This supports the definition of a circumcenter. It appears from above that the distance of the Medial triangle from the circumcenter to each point is half of that of the original triangle. This is because 2.67 cm is one-half of 5.33 cm. This is another interesting observation!

The ** incenter (I)**
of a triangle is the point on the interior of the triangle that
is equidistant from the three sides. To construct the incenter,
you take the angle bisector of the three angles of the triangle.
The Incenter of the original triangle

Next, we want to look at the incenter of the Medial triangle as well and compare the two incenters. See below for the graph.

After comparing the incenters and their distances to the points of the two triangles, I notice the same ratio exists as in the above previous cases. The distance from the medial triangle's incenter to each point is one-half that of the distance from the original triangle's incenter to each similar point.

For example, remember that point A and point E are similar points. The distance from the incenter of the medial triangle to the point E is one-half that of the distance from the incenter of the original triangle to the point A. This is true for the other points as well.

This concludes the exploration. Let's summarize!

In conclusion, if you are given a triangle, you can construct the midpoints of this triangle which just simply divide each segment of the triangle into two equal segments. Then, after connecting these midpoints, a smaller triangle is formed inside the original triangle and is known as the

This concludes the investigation
of the ** MEDIAL
TRIANGLE**.