Given three line segments j,k, and m. If these are the medians of a triangle, construct the triangle. Show that your construction is correct and that the triangle is unique.

The medians of a triangle intersect (2/3)rds of the way down, where the centroid is closer to the midpoints of each segment. So, we can trisect each median and let m be our starting median with point C our centroid.

 

Now extend the median line m.

 

Create point P along the extended median line for m that is 1/3 of the length of the median m from the end of the median m that O is closest to.

 

Create circle P' that is centered at P and has a radius of 2/3 the length of median k.

 

Now construct circle O' centered at O with radius 2/3 of the length of median j. Label one of the intersections of P' and O' as point A. This is one of the vertices of the triangle.

We can now extend segment OA.

We can construct median j by adding 1/2 of the length of OA onto the OA extended line on O away from A.

Let A' be the intersection of the extended j median line and the circle O'.

Construct a circle that is centered at A' with radious of 2/3 the length of median m. Construct another circle at O with radius 2/3 of the length of the k segment. Let point S be the intersection of these two circles.

Segment SO is 2/3rds the median k but median s is constructible.

Finally, we can connect points A, I, and S to form the triangle.

To clean it up a bit:

 

This triangle is unique because if we had chosen any of the other intersections with the circles, we would not have formed a triangle. Thus, only one triangle could have been formed with these medians.