The "middle" of a triangle, Method #1:The centroid

If we have a line segment, the midpoint is a simply defined thing, the point that cuts the segment into two equal parts. In this picture of the triangle ABC, the unlabeled red dots on each of the blue lines are the midpoints and the tick marks are place to indicate that each line segment has been cut in half.  The segment BC has two copies of a single tick mark |, the segment AC has double tick marks // and the segment AB has triple tick marks \\\.

When we talk about the center of a traingle, there are several different ways to discuss it. One of the easiest is the centroid, which is here labeled G.  The method used here is to draw a line from each vertex to the midpoint of the opposite side.  You only actually have to draw two, because the third line is promised to cross at the same place.

You will notice that the orange line segments cut the blue triangle into six parts.  Because G is the centroid, the areas of triangles AGB, BGC and AGC are all equal. More than that, each of the smaller six triangles has exactly one sixth of the area of ABC.

The method above has no coordinate system tied to it. Instead, it is planar geometry done in the classic Greek style, with diagrams that can be drawn using only a straightedge and a compass.

In the drawing here, the three vertices are given coordinates, specifically (1, 2), (3, 4) and (5, 0).  The coordinates make drawing the lines from vertices to midpoints unnecessary, because instead we can just take the average of the x coordinates and the average of the y coordinates as our corresponding x and y values.  In this case, (1+3+5)/3 = 9/3 = 3, and (2+4+0)/3 = 6/3 = 2, so the centroid is (3, 2), the point marked in red.

Tomorrow, we will look at the definition of the circumcenter, a point that is the same distance away from all three of the vertices.