Constructing a Triangle Given the Medians.
By Leighton McIntyre
Goal : To construct a triangle given the medians.
Assignment 6
The medians of a triangle intersect at the centroid (G) of the triangle. the centroid G occurs at exactly 2/3 of the length of each of the medians. The method of constructing the original triangle from the medians is by trisecting each of the medians and then finding the common intersections along 2/3 of the length of each median.
Step 1. Given any three line segments, j, k and l, construct a trisection of each of the line segments. Construct a line parallel to one of the original medians, we have chosen line segment l. Construct a copy of line segment l such that this copy intersects line segment j at 2/3 of it. Construct two circles and then a a line segment from the top vertex of the triangle formed by the original medians to the bottom of the new line segment parallel to l.
Construct a parallel line to the median line segment k, and then construct a line of the same length of k that passes through the intersection of the median line segment j and the copy of the median line segment l.
Mark the endpoint of the copy of the line segment j and then construct a line segment from the bottom of the copy of new line we label p to the end of the copy of median line segment k. Label the line q.
Finally, construct a line segment from the top of lie p to the same point of the end of copy of median line k to complete the triangle and label the line r. Note that the median line segment j, the copy of median line segment k, and the copy of median line segment l all cross in the centroid of the new triangle formed by the lines p, q and r.
Now let us clean it up a bit. Lines j, k and l are the given medians, each which we trisected and formed into a triangle. After a series of constructions we have lines p, q and r which has medians of line segment j and exact constructed copies of line segments k and l. This triangle is unique because varying lengths of medians will result in different triangles.