By: Lacy Gainey

## The CENTROID (G) of a triangle is the common intersection of the three medians.

## A median of a triangle is the segment from a vertex to the midpoint of the opposite side. The medians divide the triangle into six small triangles.

## We want to prove that the three medians of a triangle are concurrent.

We also want to show that the centroid is located ⅔ the distance from a vertex to the midpoint of the opposite side.

## Observe the triangle below,

Draw a segment between point D and E.

We can see that .

Using the Side-Angle-Side theorem, we can see that ΔABC ≈ ΔDBE. (1)

## We know since ΔABC and ΔDBE are similar.

Additionally, DE and AC are parallel.

since DE and AC are parallel.

Also,

Hence, ΔDGE ≈ ΔAGC. (2)## Observe,

We can use similar reasoning to show BG is ⅔ the length of BF.

Additionally, the G mentioned in all of these proofs is the same, showing that the medians of the triangle are concurrent.

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