Nine-point center equals a vertex IV

Another characterization of the triangle whose nine-point center coincides with one of its vertices, namely $30^{\circ}-30^{\circ}-120^{\circ}$

-triangle, is discussed here.

In $\triangle ABC$

, let $\angle A=\angle C=30^{\circ}$

and $\angle B=120^{\circ}$

. PROVE that $\triangle ACH$

is equilateral, where $H$

is the orthocenter.

See the pdf.

In $\triangle ABC$

, suppose that $\triangle ACH$

is equilateral, where $H$

is the orthocenter. PROVE that $\angle A=\angle C=30^{\circ}$

and $\angle B=120^{\circ}$

Same link as above: pdf.

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:

$a=c=R$
$\angle A=\angle C=30^{\circ}$
$\triangle ACH$ is equilateral
the reflection of $B$ over $AC$ is $O$
the reflection of $A$ over $BC$ is $H$ and the reflection of $C$ over $AB$ is $H$

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. $B=N$
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. radius $OA$ is parallel to side $CB$ and radius $OC$ is parallel to side $AB$
9. the reflection of $A$ over $BC$ is $H$ and the reflection of $C$ over $AB$ is $H$
10. $AF_a$ 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.)