<Tri-2025-2188>, hint

Since ABC is equilateral and <PMB=30, <PBM=60 

PB/BM=1/2.  AP/AN=2/1 .Hence PMN is equilateral

Let O be the circumcenter of triangle ABC and rotate  such that AC is perpendicular to A'C'. Then A'B'C' is congruent to ABC  and NP//A'C', PM//A'B', MN//B'C'.

Triangles A'B'C' and PMN are similar.