Constructing a triangle from given medians is difficult unless a firm understanding is established regarding the general aspects of the medians of a triangle. A median of a triangle is a segment from a vertex to the midpoint of the opposite side. The intersection of all medians is called the centroid of a triangle. Click here for a GSP construction of the centroid of a triangle, or click here for a detailed script of how a centroid is constructed.
After constructing a triangle with medians, it is fairly simple to construct another new triangle with the medians as the sides of the new triangle as shown below. Click here to manipulate the GSP construction below.
To construct these triangles without using GSP, follow the instructions below:
1) Construct points A, B, and C
2) Construct [m], the segment between point C and point B.
3) Construct [n], the segment between point A and point C.
4) Construct [o], the segment between point B and point A.
5) Construct [D], the midpoint f segment [o].
6) Construct [E], the midpoint of segment [n].
7) Construct [F], the midpoint of segment [m].
8) Construct [p], the segment between point A and midpoint F.
9) Construct [q], the segment between midpoint [D] and point C
10) Construct [r], the segment between point B and midpoint [E].
11) Construct line [t], parallel to segment [q] through point A.
12) Construct line [u], parallel to segment [r] through point A.
13) Construct circle [c2] centered at point A with radius length [r].
14) Construct [G], the intersection of circle [c2] and line [u].
15) Construct line [w], parallel to segment [p] through point [G].
16) Construct [H], the intersection of line [t] and line [w].
17) Construct [x], the segment between point [H] and midpoint [f].
18) Construct [p1], the polygon interior with vertices E, D, and A.
19) Construct [y], the segment between point [H] and point [A].
20) Construct [p2], the polygon interior with vertices A, B, and C.
Once this construction is mastered, it is fairly easy to construct a triangle given the medians of another triangle. The script of this is quite lengthy, so you can click here to view the script, or click here to manipulate the GSP construction below and show all of the hidden lines.
