# Barycenter

## Barycenter

The barycenter of a triangle is the point where the triangle's medians intersect. The median is a line segment joining a vertex to the midpoint of the opposite side.

## Triangle barycenter

• Enable the tool POLYGON (Window 5) and click on three different places to form a triangle. In order to close the triangle click on the first point again. Naturally, the points cannot be aligned. A triangle with vertices on points a, b and c, will be drawn as the previous figure.
• Enable the tool MIDPOINT OR CENTER (Window 2) and click on side c. A point D will appear.
• Enable the tool SEGMENT tool (Window 3), click on point C and then on point D. A segment d will appear. This segment d is called a median.
• Enable the tool MIDPOINT OR CENTER (Window 2) again and click on side a. A point E will appear. E
• Enable the tool SEGMENT tool (Window 3), click on point A and then on point E.
• Enable the tool INTERSECT (Window 2), click on the segment d and then on segment e. A point f will appear. Will the next median also pass through F? In order to observe it, enable the tool MIDPOINT OR CENTER (Window 2) again and click on side b. A point G will appear.
• Enable the tool SEGMENT tool (Window 3), click on point B and then on point G. This new drawn segment is another median. If all the procedures were followed correctly, you will have a figure similar to the one below. Point F is the barycenter. From this moment vertex F will be called barycenter. In order to do this, right click the mouse on point F and select the option RENAME. In the new window that will appear, type barycenter and click OK.

## Analysis 1

Use the tool DISTANCE OR LENGTH (Window 8). Measure the distances from point A to E and from point A to the barycenter. What do you notice? Measure the distance from the barycenter to point E. What do you observe?

## Property

The three medians of a triangle intersect at the same point that divides each median into two parts, so that the part containing the vertex is twice the length of the other one.