The problem in the image appears to involve solving a geometric problem related to a triangle, where different values for angles and lengths are given.

Let's analyze the text more closely:

1. Triangle ABC with a few points and angles provided:
   • $m(\widehat{ABC}) = 57^\circ$
   • $m(\widehat{BCA}) = 34^\circ$
   • $BD = 14$
   • $BD = 14$, $CD = 10$, $AB = AD = 18$, $\text{calculate BC}$.

It seems like we're dealing with some triangle properties and applying the triangle's rules to find side lengths or angles. I can assist with breaking down the solution step-by-step. Here's how we could approach it:

1. Use the Angle-Sum Property to calculate $m(\widehat{A})$.
2. Apply the Law of Sines or Cosines to solve for the unknown sides or angles.
3. Make use of any given side relationships or special properties in the triangle configuration.

Would you like me to guide you through these steps more explicitly or give a detailed solution?