Nine-Point Circle

 

By Courtney Cody

 

 

In this assignment, we will look at the construction of the nine-point circle and locate the center, N.  In order to understand what we are going to construct, we must begin with a good definition of nine-point circle before we can begin our construction.

 

By definition…

Š      The nine-point circle for any triangle passes through the three midpoints of the sides, the three feet of the altitudes, and the three midpoints of the segments from the respective vertices to the orthocenter.

 

We’ll begin our construction with an arbitrary triangle ABC

 

 

Since by definition the nine point circle is suppose to pass through the three midpoints of the sides of a given triangle, we will use the capabilities of Geometer’s Sketchpad (GSP) to construct the midpoints of segments AB, BC, and CA and call them D, E, and F, respectively.

 

 

Next, the definition requires that we construct the three feet of the altitudes.  By definition, an altitude is a line that extends from one vertex of a triangle perpendicular to the opposite side.  Using GSP, we will construct the three altitudes of our triangle and call the points where each of the altitudes intersects each side G, I, and J.  Additionally, the intersection of the three altitudes is called the orthocenter and is labeled point H, by convention.

 

 

If we recall, the third component of the definition of nine-point circle asks that we find the three midpoints of the segments from the respective vertices to the orthocenter.  In order to do this, we first need to find these three segments, which are pictured on the diagram in dark green.  The midpoint of each of these segments is also pictured and we call them K, L, and M.

 

 

 

At this time, we have located the nine points of the nine-point circle.  Now, we need to find the center, N.  One way to find the center of the nine point circle is to first find the circumcenter O, which is the intersection of the three perpendicular bisectors of triangle ABC.  Next, we find the Euler line, which is the line passing through the orthocenter H and the circumcenter O of our triangle ABC.  This line is pictured in dark red.  The midpoint of the Euler line is the center of the nine-point circle, called N, as desired

 

 

Lastly, we arrive at the complete illustration of our nine-point circle….

 

Suppose our original triangle had a different shape.  Would this change the appearance of the nine-point circle?  The triangle we assumed at the beginning of our construction was an acute triangle, meaning all three angles of the triangle were acute.  When this is the case, three of the nine points lie inside the triangle while the other six lie on the triangle. Specifically, points L, K, and M are inside the triangle.  Now, suppose we change the triangle such that triangle ABC becomes obtuse.  Then we notice that the same three of the nine points that were previously interior to the triangle move to the exterior of the triangle.

 

 

When triangle ABC is a right triangle, all nine points of the nine-point circle lie on the triangle, which is illustrated below:

 

Click here to see for yourself how the nine-point circle change as triangle ABC is moved around.

 


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