Advanced Triangle Centers: Centroid

Drawing a Geometry Diagram : Constructing the Centroid

You're solving a geometry problem. You start drawing the diagram, and then found out that your diagram is awful, messy, or inaccurate. You get frustrated and after an hour, a cleaner diagram is finally drawn.

Often, drawing a diagram is vital before solving any geometry problems.

I'll show how to draw the centroid with a ruler, pencil and compass.

Step 1 : Draw a nice, big, triangle.

Step 2 : Use a ruler to measure where the midpoint should be. (usually an approximate one). Then, plot them on the triangle. Do it for two of the sides.

Step 3 : Join the midpoints to the opposite vertex. Their intersection is the centroid.

This should solve your trouble for constructing centroids by hand.

The median is the line connecting a vertex of a triangle to the midpoint of the opposite side.

Here, D,E,F are the midpoints of the sides of ABC.

Notice medians $AD,BE,CF$ are concurrent.

We assume $AD,BE$ intersect at G and $CX$ goes through G for some point X on AB. Then we will try to show $X=F$. We can use mass points technique, to eventually find $BX=AX$. Thus, X is the midpoint of AB and the three medians are concurrent.

(Ceva's Theorem also gives us a trivial proof of this)

Another interesting property of medians and midpoints is that $DEF$ forms the medial triangle . It has sides parallel to those of ABC, and each side is half the length. For example, $DF = 0.5 AC, DE = 0.5 AB$. This is useful in many problems.