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# Nine-point center equals a vertex IV

Another characterization of the triangle whose nine-point center coincides with one of its vertices, namely -triangle, is discussed here.
In , let and . PROVE that is equilateral, where is the orthocenter.

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In , suppose that is equilateral, where is the orthocenter. PROVE that and .

## 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 reflection of over is and the reflection of over is