1. Construct any triangle ABC. 2. Construct the Orthocenter H of triangle ABC. 3. Construct the Orthocenter of triangle HBC. (it is A) 4. Construct the Orthocenter of triangle HAB. (it is C) 5. Construct the Orthocenter of triangle HAC. (it is B)
Proof that Orthocenter of triangle HBC is A:
By construction AH is perpendicular to BC, AB is perpendicular to CH, and AC is perpendicular to BH, where BC, CH, and BH are the three sides of triangle HBC.
So, the orthocenter of HBC must lie on AH, AB, and AC.
Therefore, the intersection of these three lines is the orthocenter, A.
A similar proof follows for HAB and HAC.
6. Construct the Circumcircles of triangles ABC, HBC, HAB, and HAC.
7. Construct the Nine Point Circles of triangles ABC, HBC, HAB, and HAC.
By definition, the nine point circle bisects any line from the orthocenter to a point on the circumcircle. [from Math World]
Also, the radius of the nine point circle is one half of the circumradius of the reference triangle. [from Math World]
Using GSP we can utilize the script tool from my assignment 5 to see the nine point circles of these four triangles. They seem to all be the same.
By dilating each of the circumcircles with its corresponding orthocenter by a scale factor of 1/2, you will notice that the circle that is created lies on the nine point circle.