From Jim Wilson’s website: 14. Prove that the three medians of a triangle are concurrent and that the point of concurrence, the centroid, is two-thirds the distance from each vertex to the opposite side.
How would you use GSP to help students understand this relationship of the triangle and its medians? How would you develop a sense of proof of the relationship with students?
We will attempt to prove that the medians of a triangle, ABC, are concurrent by using coordinate geometry. We will set our origin at A, and side AB of the triangle will lie on the x-Axis.
A has coordinates (0,0). B has coordinates (b, 0). C has coordinates (m , n).
D is the midpoint of CB and has coordinates: The Median AD has
The Median AD lies on the line: .
E is the midpoint of AC and has coordinates: The median BE has slope:
The median BE lies on the line:
F is the midpoint of AB and has coordinates: The median CF has slope:
The median CF lies on the line: .
We will now find the intersection of the lines containing medians AD and BE. and give . Solving for x yields . So centroid1 has coordinates: .
The intersection of the lines containing medians AD and CF can be found by using and .
Solving for x yields . So centroid2 has coordinates:
The intersection of lines containing medians BE and CF can be found by solving .
Solving for x gives . So centroid3 has coordinates: .
Since the three lines containing the medians have the same point of intersection, the lines containing the medians are concurrent.
(Note: Much of the symbol pushing for solving the equations was done by using the Computer Algebra System (CAS) on the TI-89. A classroom demonstration using a CAS might be a quick and convincing way to prove the concurrency of medians to high school students without getting bogged down in symbol manipulation.)
Show that the centroid lies 2/3 of the distance from the vertex to the midpoint of the opposite side:
Let’s consider the median AD. D has coordinates . The centroid has coordinates . By similar triangles, we
can take the ratio of the abscissas and the ratio of the ordinates and see that both are equal to 2/3 as desired.