Home > Affine Geometry, Geometry > Five points on a circle, centroids and perpendiculars

Five points on a circle, centroids and perpendiculars


Suppose you have {5} different points on a circle. For each triangle formed by three of these points, consider the line passing through its centroid, perpendicular to the line determined by the other two.

Proof: Pick one point {P} among the five points on the circle, and consider the homothety of center {P} and ratio {3/2}. This homothety maps the centroid of a triangle {PAB} on the midpoint of {AB}. Thus, if {A,B,C,D} are the remaining vertices, the line passing through the centroid of {AB}, perpendicular on {CD} is mapped to a line passing through the midpoint of {AB}, perpendicular on {CD}.

Now, we have the following result: Suppose you have {4} points {A,B,C,D} on a circle. Draw the lines which pass through the midpoint of a segment determined by these, perpendicular to the line determined by the other two. Then these {6} lines are concurrent. Furthermore, if {X} is the point of concurrency, {O} the center of the circle, and {G} the centroid of {ABCD}, then {G} is the midpoint of {OX}.

To prove this secondary result, note that if we take the midpoints {M,N} of {AB} and {CD}, for example, and consider the intersection {X} point of the perpendicular from {M} to {CD} and the perpendicular from {N} to {AB}, then {MXNO} is a parallelogram. Thus, the midpoint of {OX} is the same as the midpoint of {MN}, which is the centroid of {ABCD}.

Thus, among the {10} lines, there are {5} families of six lines which are concurrent. Each such two families have at least two common lines, which means the intersection point is unique.

Take a look at this MathOverflow discussion, where you can find more informations and generalizations. Here you can find a Geogebra interactive picture.

Advertisements
Categories: Affine Geometry, Geometry Tags: ,
  1. No comments yet.
  1. No trackbacks yet.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: