In my geometry class, we have seen a proof of the Euler line and one of the nine point circle theorem that were quite burdensome, involving rectangles and concyclic points and what not.
We then saw another proof of the Euler line involving homotheties, which was pretty straightforward and easy to understand. However, my teacher mentioned that he could not find a proof of the nine point circle that would be done using homotheties, thus simplifying the proof.
I then checked online and found the second proof of existence on the AOPS, and I must say I do not follow.
"We know that the reflection of the orthocenter about the sides and about the midpoints of the triangle's sides lie on the circumcircle. Thus, consider the homothety centered at H with ratio 1/2. It maps the circumcircle of triangle ABC to the nine point circle, and the vertices of the triangle to its Euler points. Hence proved."
I could see on Geogebra that the homothety centered at H of ratio 1/2 of the circumcircle gives you the nine point circle. I would then like to show that the 9 points are on the same circle centered at the middle of OH.
I am able to show that some of the points are on the circle (for example, the same homothety of the point A, a vertex of the triangle, gives you K, where K is the midpoint of AH, because AH = (1/2)KH by hypothesis), but then I am stuck when it comes to the other points and the fact about the center of the circle being the midpoint of OH, and the proof on AOPS does not help.
Any help is appreciated.
I have been working on nine circles for about 6 months now and i have 93% at the moment. Even though in practice i always beat the last cube in one attempt for some reason when ever i get there i always die. I have now died there 20+ times and i am about to give up but i was wondering if you have any tips or motivation to keep me going.
Open to all thoughts.
