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:

