Triangle Median Explorations with Geometer’s Sketchpad

By Donna Greenwood

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Given line segments j, k, m.If these are the medians of a triangle, construct the triangle. 


 

Click here for a sketch to manipulate. You can manipulate the medians, j, k and m to see how it impacts the constructed triangle ABC. This is the “backwards” version of creating a triangle of medians for any given triangle, so that’s the approach I took for the construction.


 
 

Steps to a Triangle Construction

1.  Replicate one of the medians (m in this sketch) by drawing a parallel line to segment m through an arbitrary point (C). Construct a circle with radius m at C. One of the intersections of the line with the circle and point C determine the endpoints of m.

 
 
 

2. Create a triangle out of the given segments. At the endpoints of the new segment m, create two circles of radius j and k respectively. One of the intersections of the two circles is the third vertex of the triangle, so pick one. 

 
 

3. Now we have a triangle whose leg lengths are j, k and m. From now on, references to j, k and m are to those segments, rather than the givens. Let one of the segments of the triangle be one of the medians of the new triangle (again, m in this sketch). The vertex is at point C. Note that if we could use measurement in constructions (which we can’t, of course), then we could simply measure 2/3 of the way along this median to find out where the medians intersect. Instead, construct the centroid of this triangle of medians.

 
 

4. Create a parallel at L to segment k in the sketch and replicate the segment (BL in this sketch). The endpoint is at point B in the “new” triangle.

 
 
 

5. By the definition of a median, segment CL is 1/2 the length of the side of the triangle. To get vertex A, construct the ray CL and a circle at L with radius CL. The intersection is the third vertex of the triangle, point A.

 
 

6. Connect up the points, take measurements and it works!

 
 
 

Observations and Final Notes

Why does this work? There are a couple of observations made and used in the construction that haven’t been proven. The first: Is point L, the centroid of the triangle of medians, really the midpoint of side AC of DABC? This was the foundation of the construction. Here’s a sketch:

 

Point L was constructed as the centroid of the triangle of medians, DGEC.LB was constructed as the line parallel to segment k (FG in this sketch). We let m be a median of the constructed, giving point G as the midpoint of side AB of the constructed triangle. So, by triangle mid-segment theorem, DALB is similar to DAFG, and the similarity ratio is 1/2, so AF = 1/2 AL, or AF = FL. Since L is the centroid of DGEC, FL (=AF) = 1/3 FC, and LC = 2/3 of FC, and L is the midpoint of AC.
The second observation/question is: why does creating parallel line segments “work”? See the sketch:


 
 

The answer is basically the same as for the first question. Constructing the parallels creates similar triangles. In this sketch, look at DGEC and DOHC. The measure of angle EGC = the measure of angle HOC.  The other angles aren’t marked, but you can visually follow the sets of similar triangles to see the congruent angles.