By: Tim Lehman

**Exporation 1: Dividing a triangle into 3
triangles of equal area**

Having trouble trisecting a side of a triangle?
**Click here**.

**Method II**: In
many ways, method II is simply a variation of method I. After
trisecting one side of a triangle, construct a segment connecting
the opposite vertex (point A) to only one of the points trisecting
the side (I). We then have two triangles. One (ABI) is equal to
1/3 the area of the original triangle (ABC). The other triangle
(AIC) is 2/3 the area of the original triangle (ABC).**Click
here** for a GSP sketch of the below.

To trisect the original triangle, we need to divide the larger triangle (AIC) into two equal triangles. This can be accomplished by finding the midpoint of any side of the triangle and constucting the segment from them to the opposite vertex. The two possibilities can be seen below.

**Method III:**

Another way to divide a triangle into three triangles of equal area is to find the centroid (the point where the medians intersect). After determining the centroid (point G below), construct the segments connecting the vertices to the centroid. The three triangle created are of equal area.

**Click here** for a GSP sketch of the above triangle.

*What are some other ways to divide the
triangle into three equal triangles?*

**Method I:**

This method follows directly from method III
above. The sides of the three equal sections are formed by connecting
the midpoints of the three sides of the original triangle to the
centroid. **The reasoning of why it works**
is the same as in method III.

**Click here** for a GSP sketch of the above diagram.

**Method II:**

Another method of dividing the triangle into
three equal portions includes dividing it into one triangle and
two trapezoids. First, we will find the triangle that has an area
of 1/3 the original triangle and has a side parallel to a side
of the given triangle. **How? **Then,
we divide the remaining area in half with a segment parallel to
the side of the smaller triangle. **Where
would this segment be? **We now have two trapezoids and
a triangle that are each 1/3 of the area of the original triangle.

**Click here** for a GSP sketch of a triangle trisected using this
method. (This sketch contains all construction lines as well.)

A variation of this method would be to divide the bottom two thirds in two other trapezoids. Let Q be the midpoint of LM and R be the midpoint of BC. The two trapezoids would have the same height and both bases would be the same length. Thus, the two would have the same area.

*What are other ways to divide a triangle
into three equal pieces?*