### Rachael Brown

The Problem:

Given three line segments as the three medians of the triangle. Construct the triangle.

Click here to open a GSP file that has the problem in it for you to try.

My Solution:

1. Construct the triangle that has the three medians as its sides.
2. Construct the centroid of this triangle.
3. Construct a circle whose center is the centroid and whose radius is the distance from the centroid to one of the vertices (you pick). This is what your construction looks like so far:
4. Extend the median whose vertex you used to determine the circle into a line. The intersection of the median and the circle is the midpoint of one the sides of the triangle you are building. The vertex of the blue and red side is also a point on the triangle you are building. Use this fact to create one of the sides of your triangle. Your construction should now look something like this: (the black segment is one of the sides of your triangle)
5. To get another side we need to look at the median through the vertex of the blue and ride side of our median triangle. The length of our median is actually 3/4 of the size of the side we want. So, we need to construct a circle whose center is the midpoint of the purple side and it passes through the centroid of our median triangle. Then extend the median. The intersection of the median and the circle is the third vertex of the triangle we're building. Here is a picture of it:
6. Now, we just need to connect the third vertex to the black segment we already have for the side of the triangle we are building. Here it is:
7. Click here to open a GSP file with the above picture in it. Use it to measure the medians and make sure that we constructed the correct triangle.