1. **Problem statement:** Find the measure of angle $\angle LNP$ formed by the intersection of chord $LM$ and secant $PN$ at point $N$ outside the circle. 2. **Relevant theorem:** The measure of an angle formed outside a circle by two secants, two tangents, or a secant and a tangent is half the difference of the measures of the intercepted arcs. 3. **Identify arcs:** The angle $\angle LNP$ intercepts arcs $LM$ and $MP$ on the circle. 4. **Given:** $m\angle L = 75^\circ$ and $m\angle M = 19^\circ$ are inscribed angles subtending arcs $LP$ and $MP$ respectively. 5. **Calculate arcs:** Since an inscribed angle measures half its intercepted arc, $$m\overset{\frown}{LP} = 2 \times 75 = 150^\circ$$ $$m\overset{\frown}{MP} = 2 \times 19 = 38^\circ$$ 6. **Find arc $LM$:** The circle is $360^\circ$, so $$m\overset{\frown}{LM} = 360 - (150 + 38) = 172^\circ$$ 7. **Apply the external angle formula:** $$m\angle LNP = \frac{1}{2} |m\overset{\frown}{LM} - m\overset{\frown}{MP}| = \frac{1}{2} |172 - 38| = \frac{1}{2} \times 134 = 67^\circ$$ **Final answer:** $$m\angle LNP = 67^\circ$$