<Tri-2025-2212>, hint

Triangles ABC, EAC and DBA are similar

MN//BC -->AM/ME=AN/ND,  Since CN is bisector of <ACD .  AN/ND=AC/CD

BD/AC=BA/BC=BD/c=c/a, BD=c^2/a. Similarly BE=a-BD=a-c^2/a=(a^2-c^2)/a

Finally  (c-b)(a^2+bc)=0. Hence b=c AC=AB, ABC is isosceles.