Question 51
On the small arc $AB$ of the circumcircle of the equilateral triangle $ABC$ we consider a point $N$ such that the length of the arc $NB$ is $30^\circ$. Consider the perpendicular lines from $N$ to $AC$ and $AB$, respectively. These lines intersect again the circumcircle of the triangle $ABC$ in points $M$ and $I$, respectively.
a) Prove that $IMN$ is an equilateral triangle.
b) If $H_1, H_2$, and $H_3$ are the orthocenters of the triangles $NAB, IBC$, and $CAM$, respectively, prove that $H_1H_2H_3$ is an equilateral triangle.
a) Prove that $IMN$ is an equilateral triangle.
b) If $H_1, H_2$, and $H_3$ are the orthocenters of the triangles $NAB, IBC$, and $CAM$, respectively, prove that $H_1H_2H_3$ is an equilateral triangle.