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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. 2. 3. is equilateral
4. the reflection of over is 5. the reflection of over is and the reflection of over is • (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. 2. 3. 4. is equilateral
5. the reflection of over is 6. the area satisfies 7. the circle with diameter passes through vertices and 8. radius is parallel to side and radius is parallel to side 9. the reflection of over is and the reflection of over is 10. is the geometric mean of and , and is the geometric mean of and . ( and are the feet of the altitudes from and , respectively.)

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