The Nine Point Circle Construction

The nine point circle is comprised of points of differing constructions.
First are the three mid points of triangle ABC. (green points)
Second are the feet of the altitudes of triangle ABC. (blue points)
Lastly the midpoints of the segments between the orthocenter and each vertex. (red points)
The center of the nine point circle is the midpoint from the orthocenter and the circumcenter.

With exploration one is quickly able to see that the area of the nine point circle has a constant ratio to the circumcircle of triangle ABC. The proof of this is very straight forward.

If we call the orthocenter point O then we can see the three triangles AOC, BOC, and AOB. Notice the red segments joining points Q, S, and R , the midpoints of CO, AO, and BO. This makes the red segments one half the length of the corresponding sides of triangle ABC. Thus by the Side-side-side Similarity Theorem the red triangle QRS is similar to the black triangle ABC, with a ratio of 1:2. Therefore the ratio of the area of triangle QRS to triangle ABC is 1:4 as the display below clearly shows.

Now that we have clearly shown that the areas have a ratio of 1:4 we can then look at the radii of the nine point circle and the circumcircle.

The heavy red segment is the radius of the nine point circle and the heavy blue segment is the radius of the circumcircle. We look at the triangle created by the orthocenter, circumcenter, and point B, noticing that point S is the mid point of line segment OB; also the center of the nine point circle is the midpoint of the line segment connecting the circumcenter and the orthocenter. This shows the radius of the nine point circle to be one half the length of the radius of the circumcircle. Therefore the ratio of the area of the nine point circle to the area of the circumcircle is again 1:4 as the display below clearly shows.

This proves that the area of the nine point circle of a given triangle is one-fourth the the area of its circumcircle.
To see an animation of a nine point circle in motion click here.