Investigation of a triangle formed by the medians of a given original triangle
Hee Jung Kim
A median of a triangle is a line segment connecting a vertex of the triangle with the midpoint of the opposite side of the vertex. There are a couple of properties related to the median.
1. Three medians intersect at a single point regardless of the kind of the triangle. The point where they meet is called the centroid of the triangle. The centroid divides each median in the ratio 1:2. See the proof.
2. Each median divides the triangle into two smaller triangle which have the same area. See GSP.
3. The medians divide the triangle into six smaller triangles inside the given triangle. These triangles have the same area. That is why the intersection point is called the (area) centroid as the center of gravity. See GSP.
Let's construct a second triangle with three sides having the lengths of the three medians from the first triangle.
Step 1. Construct three medians if the original triangle.
Step 2. Construct a parallel line passing through the midpont P parallel with the median RC.
Step3. Construct a line segment R'P which has the same length as the length of the median RC, where R' is an intersection point between the circle at center P with radius the length of RC.
Step 4. In the same manner, construct AR'.
Step5. We constructed a second triangle AR'P with three medians of the original triangle ABC.
Let's find some relationship between the original triangle ABC and the constructed triangle AR'P.
1. The ratio of the area of the original triangle to the constructed triangle is constant (1.3333).
2. The parameter of the original triangle is greater than the constructed triangle. Therefore the ratio of the parameter of the original triangle to the constructed triangle is greater than 1.
3. They are neither congruent nor similar.
I show a simple visual proof of the theorem that the ratio of the area of the original triangle to the constructed triangle is constant. (For the proof we should define a mid-segment of a triangle by a segment of joining midpoints of two sides of the triangle and prove that the mid-segment of a triangle is parallel to the third side and is congruent to one half of the third side. Moreover, four triangles formed by three midsegments of the triangles are all congruent.)