1. **Stating the problem:** Given a triangle with points P, L, M, and a point N inside the triangle such that $PN = NM$, we want to analyze the relationship and possibly find lengths or prove properties related to these segments. 2. **Understanding the problem:** The condition $PN = NM$ means point N lies on segment PM such that it divides PM into two equal parts, so N is the midpoint of PM. 3. **Using midpoint properties:** If N is the midpoint of PM, then by definition: $$PN = NM = \frac{PM}{2}$$ 4. **Implications:** This can be used to find coordinates or lengths if coordinates of P and M are known, or to prove congruence or similarity in triangles involving these points. 5. **Summary:** The key formula here is the midpoint property: $$PN = NM$$ which implies $$N = \text{midpoint of } PM$$ This is the main conclusion from the given information.