This is a paragraph.

Geometric properties involving medians

Let’s start with a small piece of art that is quite pleasing to look at; afterward we move on to the math:

You feel that? No doubt.

Apart from being a fine art, the above diagram will play a main part in the path we’re going to chart in today’s task. Got all of that?

Example 1

In the diagram above, are the three medians of the original triangle ; are the respective midpoints of these medians. Find the coordinates of the points given

By definition, are the midpoints of sides , respectively; so:

Similarly, because are the midpoints of medians , we have:

Let those six click. Let them stick.

Parallelism, collinearity, and similarity

Example 2

The midpoint of a median which originates from a given vertex and the midpoints of the two other sides that originate from the same vertex are collinear. (With reference to our diagram, points are collinear, so are and .)

Let’s prove that are collinear; the others are similar. From Example 1 above we already have the coordinates of these points, so we now use them to find slopes.

Since and have the same slope, it follows that are on the same line.

Example 3

The midpoint of a median which originates from a given vertex, and the midpoint of the midpoints of the two other sides that originate from the same vertex, coincide. (With reference to our diagram, the midpoint of median and the midpoint of line segment coincide. Similarly, the midpoint of median and the midpoint of line segment coincide; the midpoint of median and the midpoint of line segment also coincide.)

We’ve encountered this before; see Example 10 in our post on median of a triangle.

Example 4

The triangle formed by joining the midpoints of medians is similar to the original triangle. (With reference to our diagram, is similar to .)

Using the coordinates obtained in Example 1 above, we simply calculate the lengths of the sides of :

Since the lengths are proportional, the two triangles are similar.

Example 5

The triangle formed by joining the “feet” of the three medians is similar to the original triangle. (With reference to our diagram, is similar to .)

Like in the previous example, we simply calculate the lengths of the sides of and compare them with those of :

Since the side lengths are proportional, the two triangles are similar.

Example 6

The triangle formed by joining the “feet” of the three medians is similar to the triangle formed by joining the midpoints of the three medians. (With reference to our diagram, is similar to .)

Based on the calculations in Example 4 and Example 5, we saw that:

Since the side lengths of and are proportional, the two triangles are similar.

Example 7

The line joining the midpoints of two medians is parallel to the third side and equal to one-fourth of its length.

Consider medians and in our “fine-tistic” diagram. The midpoints of these medians are and , respectively. We show that line is parallel to side . This immediately follows by calculating slopes:

Since and have the same slopes, they are parallel. Furthermore, in Example 4 we saw that the length of is a quarter of the length of , so this concludes the proof.

Example 8

The line joining the midpoints of two sides of a triangle is parallel to the third side and equal to one-half of its length.

This is more familiar.

Preservation of the centroid

Example 9

The triangle formed by joining the midpoints of the three medians has the same centroid as the original triangle. (With reference to our diagram, has the same centroid as .)

In view of Example 3, note that this result is expected. But let’s prove it using the centroid formula. Since the coordinates of are , we know that the centroid of is located at the point

Next, the coordinates of points were calculated in Example 1; they are:

To find the centroid of , we add up the -coordinates and divide by , then repeat the same thing for the -coordinates. Adding the -coordinates gives

Similarly, adding the -coordinates gives . Finally we divide each of these sums by and then it follows that the centroid of is the same as that of .

Example 10

The triangle formed by joining the “feet” of the three medians has the same centroid as the original triangle. (With reference to our diagram, has the same centroid as .)

Just like we did in Example 9 above, we use the coordinates of points to find the centroid. Recall, from Example 1, that the coordinates of are:

Adding the -coordinates also gives . Therefore, the centroid of is located at , same point as that of

Takeaway

The triangle obtained by joining the midpoints of the sides of a triangle is usually called the medial triangle. The process of constructing the medial triangle can be repeated until a point that approximates the centroid (of the original triangle) is reached.

In passing, by appearing somewhat playful — especially by “touting” our “artistic” diagram — we’ve exhibited a trait that can be traced to our trade.

1. PROVE that the triangle formed by joining the midpoints of the sides of an isosceles triangle is also an isosceles triangle.
[Same for an equilateral triangle.]
2. Let be such that . Prove that the length of the median from vertex is half of the length of the hypotenuse .
3. Let be such that . Prove that the distance from the centroid to vertex is one-third of the length of the hypotenuse .
[You’ll love right triangles. They can be so nice . See Exercise 10 below.]
4. [A little “calculus”] Let be the vertices of . Define as the midpoint of the median from , namely . Define as the midpoint of and . Define to be the midpoint of and , and so on. PROVE that

[Observe that if , then we obtain the midpoint of side , as expected. In the limit , we obtain , the coordinates of . Beautiful, simple calculus.]

5. Let be the triangle obtained by joining the midpoints of the sides of . PROVE that the area of is a quarter of the area of .
6. Let be the triangle obtained by joining the midpoints of the three medians of . PROVE that the area of is the area of .
7. Find a triangle that satisfies the condition that the product of the slopes of the three sides is equal to , and the product of the slopes of the three medians is also equal to .
[An example is with coordinates at . Find a different example.]
8. Find a triangle in which the product of the slopes of the three sides is , while the product of the slopes of the three medians is .
9. PROVE that a triangle is isosceles if, and only if, two of its medians are equal in length. [Similarly, a triangle being equilateral is equivalent to having three medians that are equal in length.]
10. PROVE that the triangle formed by joining the midpoints of the sides of a right triangle is also a right triangle. Moreover, the angle of the new triangle is on the hypotenuse of the original triangle. [Also, PROVE that the triangle formed by joining the midpoints of the three medians of a right triangle is also a right triangle, and that its angle is on the hypotenuse of the right triangle obtained by joining the midpoints of the original, parent right triangle.] A picture is worth a thousand words. And then a good picture??? In order to provide a visual aid to this exercise, below is a “mouth-watering” diagram, simultaneously serving as a befitting addition — and conclusion — to the “eye-catching” one we had at the beginning of the post: