Assignment 4:

Concurrence of Medians

by

Jenny Johnson

**Medians**

A
median of a triangle is the segment from a vertex to the midpoint of the
opposite side. Each triangle will
have three medians – one segment from each of the three vertices. One
median is shown as a dotted purple segment in the triangle below.

The
CENTROID (G) of a triangle is the common intersection of the three
medians. In order to construct the
centroid, we only need to construct 2 medians and find their intersection. Since the three medians are concurrent,
the third median will intersect the other two at their intersection. However, we will construct all three
medians for instructional purposes.
Their intersection is the centroid (G).

**Proof that
medians are concurrent**

While
the picture on GeometerÕs Sketchpad demonstrates that the medians are
concurrent at a single point, it does not prove that they will always be
concurrent. Firstly, it is fairly
obvious that the medians of any triangle will intersect inside the
triangle. Since the median of any
side of the triangle will always be contained on the segment that forms the
side of the triangle, then the segment connecting that median to the opposite
vertex will also necessarily be on the interior of the triangle. The point of concurrency of the medians
of an acute triangle is pictured above.
The centroids for an obtuse and a right triangle are pictured below.

In
order to prove the concurrency of the medians, let us start with triangle
ABC. Now, let us construct E to be
the midpoint of AC and F to be the midpoint of AB. We can also construct the segments BE and CF. We know these two segments will
intersect. LetÕs call their point
of intersection G. Form segment
FE.

Segment
FE is called the midsegment of the triangle because it is the segment that
connects the midpoints of two sides of a triangle. Since it is the midsegment, we know it is parallel to the
third side BC and half the length of BC.
Since FE is parallel to BC, we know that angle EFG is congruent to angle
GCB by the alternate interior angles theorem. Also by the alternate interior angles theorem, we know that
angle FEG is congruent to angle GBC.
Since we know two angles are congruent in triangles FGE and CGB, then
the two triangles are similar by the AA Similarity Postulate. The similar
triangles are shaded in the picture below.

Since
the triangles are similar, their sides are proportional. So,

So
median CF cuts median BE at a point exactly 2/3 of the way from B to E. Median BE cuts median CF at a point
exactly 2/3 of the way from C to F.
If we were to construct the midpoint of CB (point D), then the median
from A to D must also cut BE at a point 2/3 of the way from B to E. Since weÕve already identified this
point as G, then AD must intersect the other two at point G. The diagram below shows the two similar
triangles formed when we draw ED as our midsegment.

We
have thus proven that the three medians in a triangle must be concurrent at a point
G and this point is exactly 2/3 the distance from each vertex to the midpoint
of the opposite side.

**How would
GSP help students discover this property of the centroid?**

In
order for students to make their own discoveries about the centroid, I might give
them the following sequence of constructions and questions to explore on GSP:

1.
Construct a
triangle ABC.

2.
Construct the midpoint of BC and label
it D.

3.
Construct the midpoint of AC and label
it E.

4.
Construct the midpoint of AB and label it F.

5.
Connect A to D, B to E, and C to F. What do you notice about
these three segments?

6.
Drag A to form different types of
triangles. What do you notice
about the segments?

7.
Mark the point of intersection of the
three medians G.

8.
Create segments CG and GF. Measure the length of the two
segments. Drag A to form different
triangles. What do you notice
about the lengths of the segments?

9.
Now calculate the ratio of CG to GF (go
to the Measure menu, click Calculate, enter CG ü GF). Now drag A to form different triangles. What do you notice?

10.
Repeat steps 8 and 9 for
segments BG and GE. What do you
notice?

11. Repeat
steps 8 and 9 for segments AG and GD.
What do you notice? Do you
think this will always be true?

12. How could we be
sure that G will always have the same properties youÕve noticed here? Is there another point in the triangle
ABC that could have the same properties G has? If so, what point?
If not, what does that say about G?

13. Summarize your findings
and conjectures about the medians of a triangle and the point where they cross.

**Click
here for your own exploration of medians and
centroids.**