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*:

- is equilateral
- the reflection of over is
- the nine-point circle of is the same as the incircle of
- 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*:- is equilateral
- the reflection of over is
- the area satisfies
- the circle with diameter passes through vertices and
- the nine-point circle of is the same as the incircle of
- radius is parallel to side and radius is parallel to side
- the reflection of over is and the reflection of over is
- is the geometric mean of and , and is the geometric mean of and . ( and are the feet of the altitudes from and , respectively.)