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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 30^{\circ}-30^{\circ}-120^{\circ}-triangle. pdf here.

In \triangle ABC, let \angle A=\angle C=30^{\circ} and \angle B=120^{\circ}. PROVE that the nine-point circle of \triangle ABC coincides with the incircle of \triangle ACH, where H is the orthocenter.

See the pdf here.

Suppose that the nine-point circle of ABC coincides with the incircle of \triangle ACH, where H is the orthocenter. PROVE that \angle A=\angle C=30^{\circ} and \angle B=120^{\circ}.

Same link as above: pdf here.

Takeaway

In any triangle ABC, let a,b,c be the side-lengths, H the orthocenter, O the circumcenter, and R the circumradius. Then the following statements are equivalent:

  1. a=c=R
  2. \angle A=\angle C=30^{\circ}
  3. \triangle ACH is equilateral
  4. the reflection of B over AC is O
  5. the nine-point circle of ABC is the same as the incircle of ACH
  6. the reflection of A over BC is H and the reflection of C over AB is H

Task

  • (Aufbau) In triangle ABC, let a,b,c be the side-lengths, R the circumradius, O the circumcenter, N the nine-point center, and H the orthocenter. PROVE that the following statements are equivalent:
    1. B=N
    2. a=c=R
    3. \angle A=\angle C=30^{\circ}
    4. \triangle ACH is equilateral
    5. the reflection of B over AC is O
    6. the area K satisfies K=\frac{1}{2}ac\cos A=\frac{1}{2}ac\cos C
    7. the circle with diameter OH passes through vertices A and C
    8. the nine-point circle of ABC is the same as the incircle of ACH
    9. radius OA is parallel to side CB and radius OC is parallel to side AB
    10. the reflection of A over BC is H and the reflection of C over AB is H
    11. AF_a is the geometric mean of BF_a and CF_a, and CF_c is the geometric mean of AF_c and BF_c. (F_a and F_c are the feet of the altitudes from A and C, respectively.)