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# Nine-point center equals a vertex V

A follow-up/corollary to our previous post: yet another characterization of the triangle whose nine-point center coincides with one of its vertices, namely the -triangle. pdf here.
In , let and . PROVE that the nine-point circle of coincides with the incircle of , where is the orthocenter.

See the pdf here.

Suppose that the nine-point circle of coincides with the incircle of , where is the orthocenter. PROVE that and .

Same link as above: pdf here.

## Takeaway

In any triangle , let be the side-lengths, the orthocenter, the circumcenter, and the circumradius. Then the following statements are equivalent:

1. is equilateral
2. the reflection of over is
3. the nine-point circle of is the same as the incircle of
4. the reflection of over is and the reflection of over is

## Task

• (Aufbau) In triangle , let be the side-lengths, the circumradius, the circumcenter, the nine-point center, and the orthocenter. PROVE that the following statements are equivalent:
1. is equilateral
2. the reflection of over is
3. the area satisfies
4. the circle with diameter passes through vertices and
5. the nine-point circle of is the same as the incircle of
6. radius is parallel to side and radius is parallel to side
7. the reflection of over is and the reflection of over is
8. is the geometric mean of and , and is the geometric mean of and . ( and are the feet of the altitudes from and , respectively.)