Proof of Concurrency for the Medians of a Triangle
by: Maggie Hendricks
The centroid (typically denoted by the letter G) of a triangle is the point of concurrency of the three medians. A median of a triangle is the segment from a vertex to the midpoint of the opposite side.
Click here to see a triangle with centroid constructed from connecting each of the three medians of the side lengths with its opposite vertex.
Now let’s explore some properties and characteristics when the three medians of a triangle are connected to their opposite vertices. In an attempt to prove that the three medians of a triangle are concurrent and that the distance from the median to the centroid is one-third the distance from the median to the opposite vertex we will need to go through a number of proofs that build upon each other.
We could go through a similar series of proofs using and AX and BZ to show that the same ratio of segment lengths hold true for all three medians connected to their opposite vertices. In order to then prove that the three medians are concurrent, we would need to repeat a series of proofs similar to the ones above using AX and BZ. Without this, it would only suffice to say that our point G is the same AX/BZ and AX/CY; we must prove that the point G is the same for BZ/CY in order for this point G to be the point of concurrency of our three medians.