Geometry Quickies #1
It seems that in geometry classes in school, we seldom use coordinates. Yet lots of really nice patterns pop out when we put locations on the geometric objects we are talking about. Here is a little example.
Using the Geometer’s Sketchpad or on a sheet of Graph paper, construct three points to be the vertices of a triangle. Now construct the medians to find the geocenter (also called the centroid) where they intersect, and find the coordinates of this point. Remember there is some rounding error possible on the Sketchpad, and measurement error is probable when you work by hand. Now write down the coordinates of the vertices and compare to the coordinates of the geocenter, do you see a relationship? If you’re using the sketchpad you can move the vertices to try and make all the points have integer values, that should help you see the pattern. Try several examples before you read the rest, and see if you can find the pattern on your own. In the sketchpad example below, I picked the points (2,3); (8, 2); and (5.4). Can you tell where the geocenter will be?
Did you correctly predict the point (5,3)? If so, you probably have discovered that the coordinates of the geocenter is simply the arithmetic average of the coordinates of the vertices. The x-coordinate of G is given by adding the x-coordinates of the three vertices and dividing by three. In this case (8+2+5)/3=5. The y-coordinate, likewise, is the average of the three y-values of the vertices. Lets see if we can use a little geometry and vector algebra and prove that the happy "coincidence" will always be true.
We begin by labeling three points, lets call them P1, P2 and P3,in the plane with abstract labels. We’ll call P1(x1,y1). Tricky, eh?? and P2=(x2,y2).. okay, you get the point. Now we need to find the midpoint of the segment from P1 to P2. Easy, just average the x and y values of the two endpoints, since the midpoint IS the average of the endpoints. We’ll call the midpoint M=( ). Now we need to find a vector that goes from P3 to M, to do that we just take the coordinates of P3 and subtract the coordinates of M giving us . Whew, that looks like a mess, but it will all simplify out real soon, so hang on.
Now we can use one of those wonderful geometry theorems you studied so hard to learn, to make the job easy. We know that the geocenter, the intersection of the medians, is located 2/3 of the way from the vertex, P3, to the midpoint of the opposite side, M. All we have to do, then, is start with the coordinates of P3, add 2/3 of the vector distance from P3 to M, and we’ll have captured the beast we seek.
The complete expression, and the simplification, looks like this:
Notice that both the left and right side of the point are the same except for the x, y parts, that is, if we simplify the left coordinate, the right coordinate is exactly the same with y’s in place of the x’s. Simplifying and combining the common terms gives us :
Whoa, this means that to find the Geocenter, just average the x coordinates and average the y coordinates and your done? Yes, and in the example above using (8,2), (2,3) , and (5,4); the geocenter has an x-coordinate of (8+2+5)/3=5 and a y-coordinate of (2+3+4)/3=3.
This might seem, in retrospect, to be expectable. After all, the Geocenter is the center of gravity. When we find the "center of gravity" of a line segment, the midpoint, we do the same thing, we average the x-coordinates to get the x-value of the midpoint and the y-values for y. So it is not so unusual, perhaps that it works in a triangle.