Untitled Document

The Centroid of the Quadrilateral
by
Mike Patterson

Uses of the Centroid in the Physical World

"The term centroid is of recent coinage. It is used as a substitute for the older term "center of gravity." (MT Jan 1960) The properties of the centroid show up in a number of different disciplines and research areas such as: location searches, the sound spectrum and shape properties.
The common investigation concerning the centroid for middle school students is the triangle. We will change the focus to the quadrilateral for this essay. Although our focus is the quadrilateral, the patterns and ideas found within the triangle extend naturally to the quadrilateral. The first place that this shows up is in its construction.

Construction of the Centroid of a Quadrilateral

Any quadrilateral can be divided into two triangles by drawing one of the diagonals. Find the centroids of these two triangles and then connect the line segment between them. Create two new triangles in the quadrilateral by drawing the other diagonal. Find the centroids of these two triangles and then connect the line segment between them. The two line segments of the four centroids intersect at G, the centroid of the quadrilateral, as shown in the animation.

Control Factor - Cyclic Quadrilaterals

Cyclic Quadrilaterals are the group of quadrilaterals where all four vertices are inscribed on a circle. It is also true of this type of quadrilateral that the sum of opposite angles total 180 degrees. By using a particular subgroup of the quadrilateral, I hope to find patterns that I can extend to all quadrilaterals.

Follow the Centroid

In the below diagrams I have traced the four centroids of the triangles in the quadrilateral in red, blue, green and yellow. The black trace is the centroid(G) of the quadrilateral.

Method #1 - Holding the distance between the vertices constant and studying the locus of the centroids of the triangles in the quadrilateral and the centroid of the quadrilateral.

When the distances between the vertices is held fixed the quadrilateral remains unaltered. It is simply rotated around the center point of the circle with which it is inscribed. The pattern generated is a series of concentric circles. This is of no surprise by nature of the motion in which we move the shape, but what is of interest is the order and size relationship of the circles. The black circle (G - centroid of the quadrilateral) is quite variable.

SHAPE #1 - similar angle size
Here the centroid(G) of the quadrilateral creates a circular path tight to the center of the circle. The color pattern from largest to smallest is blue, yellow, red, green, and black(G). Does this pattern hold for all cyclic quadrilaterals?......nope! SHAPE #2 - one large angle
By altering the quadrilateral we see that it dramatically changes the relationship of the paths of the centroids. Notice that the red centroid is distant from the others. This is because of the large obtuse angle. The path pattern here is red, green, blue, black(G) and yellow. SHAPE #3 - concave
By altering the quadrilateral again we see that it dramatrically changes the relationship of the paths of the centroids. The path pattern here is black(G), yellow, red, green, and blue. In this particular case the centroid(G) is outside the circle.

We do notice that the size of angles of the quadrilateral directly affect the size and order of the circles formed by tracing the centroids. If the quadrilateral was a square (diagram on the right) we see four equal circles and that the centroid of the quadrilateral(G) would be the center of the original circle.
As we alter the angles of the quadrilateral we would see that large obtuse angles spread the circles out as in SHAPE #2 above. While if the angles of the quadrilateral are quite close in size we would see a clustering as in SHAPE #1. Finally, when we alter the shape dramatically as in SHAPE #3, we find that it is possible to have the centroid of the quadrilateral be outside of the quadrilateral.

Method #2 - Holding the distance between three vertices constant and while moving the fourth one along the circumcircle.

In this section we will follow the traced path of the centroid(G) of the quadrilateral. It will become quite obvious that the path does and can extend outside of the quadrilateral. This phenomenon occurs as the quadrilateral folds onto itself creating a concave quadrilateral. In the below diagrams we are moving point D while holding C, B and A constant.

SHAPE #1 - two fixed equal lengths (CAB = A)
When we fix CAB = A and move D around the circle we find a symmetrical shape form. A shape that goes through the center of the circle and through the centroid of the triangle formed by the two fixed sides. SHAPE #2 - two unequal fixed lengths (CAB = .CAB)
When CAB is about half of A we see the figure begin to skew away from symmetry. It begins to slightly tilt away from the shorter fixed side. The intersection of the figure still occurs at the centroid of the triangle formed by the fixed sides. SHAPE #3 - two very different fixed lengths (CAB < .25AB)
When CAB is much smaller than A we see the figure begin to skew dramatically away from symmetry. It begins to curve away from the shorter fixed side. The intersection of the figure still occurs at the centroid of the triangle formed by the fixed sides.

SHAPE #4 - where CAB and A are fixed sides but A is between CAB (concave case)
When D is moved about the circumference of the circle the quadrilateral is convex for a very short period of time (when D is between A and C). Due to the predominant concave nature of this shape, its path is dramatically different from the previous examples. SHAPE #4 - continued
The path comes to a cusp at the centroid of the triangle formed by the fixed sides. No loop is formed but the path does still have a critical point at the green centroid.

Method #3 - Holding the distance between two vertices constant and while moving a fixed distance between the other two vertices.

To explain which points are moving and their relationship I will refer the first diagram below. For this method NP will be a fixed unmoving point, while TS a fixed length will be moving.

NP a fixed small length

SHAPE #1 - Two fixed equal lengths (NP = TS)
Symmetry appears as we manipulate the two fixed equal lengths. With some further investigation I am led to believe that the perpendicular bisector of NP would form the axis of symmetry. (Look below for a similar example) SHAPE #2 - Length ST = Diameter
Symmetry once again becomes evident, while also generating only two circles. Thus some of the centroids are following a identical path. (Look below for a similar example) SHAPE #3 - TNP and SPN are large obtuse angles
I saw no particular new pattern or important information through this example.

NP a fixed larger length but smaller that the diameter

SHAPE #4 - Two fixed equal lengths (NP = TS)
The change in NP did not effect the symmetry. It changed the relationship between the circle but the symmetry to the perpendicular bisector of NP still held. SHAPE #5 - Length ST = Diameter
Two circles were formed again. We also notice in the larger model that the Centroid of the quadrilateral goes through two centroids. SHAPE #6 - TNP and SPN are large obtuse angles
Again, I saw no particular new pattern or important information through this example other than the size of the path of the centroid(G) extending outside the circle.

NP a fixed length equal to the diameter

SHAPE #7 - TS is a very small length
As T and S approach the same point we see the centroids create tangents to each other. If T and S became the same point we would have a right triangle, and its centroid path would be a circle centered to the original circle's center. SHAPE #8 - TS is larger but less than the diameter
The path of G falls exactly on the path of one of the other centroids. The paths seem to hold symmetry though the perpendicular bisector of NP. SHAPE #9 - TS is very close to the length of the diameter
The path of G falls extends well beyond the boundaries of the original circle, while still hold it symmetry to the perpendicular bisector of NP.

Method #4 - Holding a single point fixed while moving three fixed points that preserve the distance between them.

To explain which points are moving and their relationship I will refer the first diagram below. For this method the point T will remain fixed while angle NPS moves around the circle.

SHAPE #1 - Angle NPS is acute
From above I have begun to see that the size of the angles of the quadrilateral make the most impact on the path of the centroid. Here we see a loop form when the quadrilateral degenerates. SHAPE #2 - Angle NPS is just less than 90 degrees
We see the loop has reduced down, and the path is straightening. SHAPE #3 - Angle NPS is a right angle.
The path of G straightens out to become a line through the center of the circle.

SHAPE #4 - Angle NPS is obtuse
From above SHAPE #5 - Angle NPS is s larger obtuse.
The larger the angle alters the path of the centroid such that a horseshoe like shape is formed SHAPE #6 - Angle NPS is very close to being straight
As angle NPS approaches 180 degrees see the paths of the centroids all becoming a single circle. The line extending outside of the circle is the portion the centroids path while the quadrilateral is degenerate.

Summary: Many hours of investigating using the "what if" approach helped me to find many interesting patterns concerning the centroid. These patterns better focus my approach but they didn't seem to reveal any startling or special relationships. This felt much like the true method of discovery where one pursues numerous paths in hopes to make that break through, not ever really knowing how far it is until the break through is made. Thus far I don't feel that I have exhausted all the possibilities, nor have I written up all that I have done....... This exploration will continue......

The end for now


SHAPE #1 - similar angle size Here the centroid(G) of the quadrilateral creates a circular path tight to the center of the circle. The color pattern from largest to smallest is blue, yellow, red, green, and black(G). Does this pattern hold for all cyclic quadrilaterals?......nope!	SHAPE #2 - one large angle By altering the quadrilateral we see that it dramatically changes the relationship of the paths of the centroids. Notice that the red centroid is distant from the others. This is because of the large obtuse angle. The path pattern here is red, green, blue, black(G) and yellow.	SHAPE #3 - concave By altering the quadrilateral again we see that it dramatrically changes the relationship of the paths of the centroids. The path pattern here is black(G), yellow, red, green, and blue. In this particular case the centroid(G) is outside the circle.
We do notice that the size of angles of the quadrilateral directly affect the size and order of the circles formed by tracing the centroids. If the quadrilateral was a square (diagram on the right) we see four equal circles and that the centroid of the quadrilateral(G) would be the center of the original circle. As we alter the angles of the quadrilateral we would see that large obtuse angles spread the circles out as in SHAPE #2 above. While if the angles of the quadrilateral are quite close in size we would see a clustering as in SHAPE #1. Finally, when we alter the shape dramatically as in SHAPE #3, we find that it is possible to have the centroid of the quadrilateral be outside of the quadrilateral.


SHAPE #1 - two fixed equal lengths (CAB = A) When we fix CAB = A and move D around the circle we find a symmetrical shape form. A shape that goes through the center of the circle and through the centroid of the triangle formed by the two fixed sides.	SHAPE #2 - two unequal fixed lengths (CAB = .CAB) When CAB is about half of A we see the figure begin to skew away from symmetry. It begins to slightly tilt away from the shorter fixed side. The intersection of the figure still occurs at the centroid of the triangle formed by the fixed sides.	SHAPE #3 - two very different fixed lengths (CAB < .25AB) When CAB is much smaller than A we see the figure begin to skew dramatically away from symmetry. It begins to curve away from the shorter fixed side. The intersection of the figure still occurs at the centroid of the triangle formed by the fixed sides.
	SHAPE #4 - where CAB and A are fixed sides but A is between CAB (concave case) When D is moved about the circumference of the circle the quadrilateral is convex for a very short period of time (when D is between A and C). Due to the predominant concave nature of this shape, its path is dramatically different from the previous examples.	SHAPE #4 - continued The path comes to a cusp at the centroid of the triangle formed by the fixed sides. No loop is formed but the path does still have a critical point at the green centroid.

NP a fixed small length

SHAPE #1 - Two fixed equal lengths (NP = TS) Symmetry appears as we manipulate the two fixed equal lengths. With some further investigation I am led to believe that the perpendicular bisector of NP would form the axis of symmetry. (Look below for a similar example)	SHAPE #2 - Length ST = Diameter Symmetry once again becomes evident, while also generating only two circles. Thus some of the centroids are following a identical path. (Look below for a similar example)	SHAPE #3 - TNP and SPN are large obtuse angles I saw no particular new pattern or important information through this example.
NP a fixed larger length but smaller that the diameter

SHAPE #4 - Two fixed equal lengths (NP = TS) The change in NP did not effect the symmetry. It changed the relationship between the circle but the symmetry to the perpendicular bisector of NP still held.	SHAPE #5 - Length ST = Diameter Two circles were formed again. We also notice in the larger model that the Centroid of the quadrilateral goes through two centroids.	SHAPE #6 - TNP and SPN are large obtuse angles Again, I saw no particular new pattern or important information through this example other than the size of the path of the centroid(G) extending outside the circle.
NP a fixed length equal to the diameter

SHAPE #7 - TS is a very small length As T and S approach the same point we see the centroids create tangents to each other. If T and S became the same point we would have a right triangle, and its centroid path would be a circle centered to the original circle's center.	SHAPE #8 - TS is larger but less than the diameter The path of G falls exactly on the path of one of the other centroids. The paths seem to hold symmetry though the perpendicular bisector of NP.	SHAPE #9 - TS is very close to the length of the diameter The path of G falls extends well beyond the boundaries of the original circle, while still hold it symmetry to the perpendicular bisector of NP.


SHAPE #1 - Angle NPS is acute From above I have begun to see that the size of the angles of the quadrilateral make the most impact on the path of the centroid. Here we see a loop form when the quadrilateral degenerates.	SHAPE #2 - Angle NPS is just less than 90 degrees We see the loop has reduced down, and the path is straightening.	SHAPE #3 - Angle NPS is a right angle. The path of G straightens out to become a line through the center of the circle.

SHAPE #4 - Angle NPS is obtuse From above	SHAPE #5 - Angle NPS is s larger obtuse. The larger the angle alters the path of the centroid such that a horseshoe like shape is formed	SHAPE #6 - Angle NPS is very close to being straight As angle NPS approaches 180 degrees see the paths of the centroids all becoming a single circle. The line extending outside of the circle is the portion the centroids path while the quadrilateral is degenerate.