Building a Triangle Given the Medians

By: Laura Lowe

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

I began by working backwards to try to find relationships.  I started by drawing a triangle and constructing its medians, and then constructed a triangle with sides congruent to the medians of the first triangle.  (Click here to see how to construct a triangle from line segments.)

I soon noticed that LEGA  LHIJ, LDGA  LIHJ, and LDGC  LIJH.  This means that the angles between the medians of a triangle are the same as the angles between the corresponding sides of the triangle constructed from the medians.   So now I knew something about the angles.

I also knew that

 and

from the properties of medians.

So now I had enough information to solve the problem: Given line segments j, k, m.  If these are the medians of a triangle, construct the triangle.

After constructing the triangle from the medians,

I trisected each side.

Then I constructed the lines IJ, IH, and a line through I, parallel to HJ.

Now I is the centroid of the triangle with j, k, and m as the medians.  (I could have done this for any of the vertices.)  Then I constructed a circle with center I and radius IP.

IQ is congruent to k, and is a median of the triangle I am building.  Next, I constructed a circle with center I and radius IK.

RL is congruent to j, and is another median of the triangle I am building.  This also allows me to construct one side of the triangle, QL.  R is also the midpoint of the second side of the triangle, so I constructed the line QR and O is the midpoint of the third side, so I constructed the line LO. The third vertex, S, of the triangle is at the intersection of QR and LO.

Clean it up a bit and we get:

And for those of you who like to see numbers: