Philippa M. Rhodes

## Write-up 8

Construct the Orthocenter (H) of triangle ABC. Construct the Nine-Point circles for triangles ABC, HBC, HAC, and HAB.
The orthocenter of a triangle is the common intersection of the three lines containing the altitudes. An altitude is a perpendicular segment from a vertex to the line of the opposite side.

Here is the orthocenter (H) of triangle ABC.

The Nine-Point circle for any triangle passes through the three mid-points of the sides, the three feet of the altitudes, and the three mid-points of the segments from the respective vertices to the orthocenter. Thus, we have the Nine-Point circle (with center N) of triangle ABC.

Next, let's overlay the Nine-Point circle of triangle HAB.

WOW! It's the same circle. Why?

Let's see how the Nine-Point circle of triangle HAC compares to the other two.

Again, it is the same circle. Why?

Finally, we see that the Nine-Point circle of triangle HBC is the same as for the previous three triangles. Once again, Why?

The question, "WHY?", can be answered by looking at the construction of the Nine-Point circle of each triangle.

To start the construction of the Nine-Point circle of triangle HAB, we will draw the orthocenter (H). Side HA and side AB must be extended to intersect the altitudes starting at vertices B and H, respectively.

Thus we have the first three points. Notice that the same three point are the feet of the altitudes of triangle ABC. Starting at the top point and moving couterclockwise, we see that he first point is the intersection of the altitude from vertex A to side BC. The next point starts at vertex B and intersects the line containing side AC, and the third point originates at vertex C and intersects the extended side AB.

The other six points are simply the three midpoints of the sides of triangle HAB and the three midpoints of the segments from the respective vertices to the orthocenter.

Again, notice the relationship between these points and the same six points of triangle ABC. Sides HA and HB of triangle HAB are the segments from vertices A and B, respectively, to the orthocenter of triangle ABC. Side AB is shared by the two triangles. Sides AC and BC of triangle ABC are the segments from vertices A and B, respectively, to the orthocenter of triangle HAB. Finally, side HC is the segment from vertex H to the orthocenter of triangle HAB, and it is the segment from vertex C to the orthocenter of triangle ABC.
Hence the midpoints of the segments are common to both triangle ABC and triangle HAB.

So, we see why the Nine-Point circle is the same for both triangle ABC and triangle HAB.

Similar comparisons can be made with triangles HAC and HBC to show why their Nine-Point circles are the same as the circles of triangle ABC or of triangle HAB.