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Centroid and median of triangle

A median is a line that goes from one corner of a triangle to the midpoint on the opposite side.

Theory

The Centroid and Medians

The medians have one common point of intersection. This point is called the centroid G. The centroid G divides all the medians in the ratio 2:1. This results in the following relationships:

AGGP=BGGQ=CGGR=2

When you want to find the centroid of a triangle, you need to draw two of the medians. You draw the medians by drawing a straight line from every corner to the midpoint on the opposite side. The intersection of these lines is the centroid.

Example 1

A triangle △ABC has the sides AB=6, AC=4 and BC=7. Construct the centroid of the triangle.

Before you construct the centroid G, you need to construct the triangle with the given measures. Start with the line AB=6. Set the compass’s radius to 7 and make a faint circle with center B. Then, set the compass’s radius to 4 and make a faint circle with center A. The corner C appears as either of the intersection points between the two circles. Then you end up with the following triangle:

Example of construction of centroid of triangle 1

Then you construct the angle bisector for two of the sides. At the intersection between these, you have the incenter, which you call I.

Then you find the midpoint to two of the sides and draw a straight line from each of these points to the opposite corner. The centroid G of the triangle is the intersection between these two lines:

Example of construction of centroid of triangle 2

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How to Find the Orthocenter and Altitudes of a Triangle