1. **State the problem:** Given circle A with points A, B, and C on the circumference, and interior point D such that $m\angle ADB = 86^\circ$, find $m\angle ACB$. 2. **Recall the theorem:** The measure of an inscribed angle ($\angle ACB$) is half the measure of the intercepted arc or the related central angle or angle formed inside the circle. 3. **Use the property of angles inside a circle:** The angle formed inside the circle by two chords ($\angle ADB$) is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. 4. **Apply the relationship:** Since $\angle ADB = 86^\circ$ is an interior angle, the inscribed angle $\angle ACB = x$ is related by: $$x = \frac{1}{2} m\angle ADB$$ 5. **Calculate $x$:** $$x = \frac{1}{2} \times 86^\circ = 43^\circ$$ 6. **Final answer:** $$m\angle ACB = 43^\circ$$