To solve this problem, let’s analyze the given information:

Given:

1. $C$ is the point of tangency.
2. $\angle BCF = 50^\circ$.
3. $m\overarc{AC} = 4 \cdot m\overarc{DE}$.
4. $m\overarc{AD} = 120^\circ$.

We are tasked with finding $\angle ABC$.

Step-by-step Solution:

1. Interpretation of the geometry:

• The circle has a tangency at $C$, meaning any tangent or chord subtended by an arc will relate to the inscribed angle theorem and tangent-secant properties.
• We know $m\overarc{AC} = 4 \cdot m\overarc{DE}$, so $m\overarc{DE}$ is a smaller segment of the circle.

2. Central angle calculation:

The central angle $\angle CAD$, subtended by $m\overarc{AD}$, is $120^\circ$. This helps determine the rest of the circle’s arc distribution.

3. Relationship between arcs and angles:

The inscribed angle $\angle ABC$ is subtended by $m\overarc{AC}$, and the property of inscribed angles states: $\angle ABC = \frac{1}{2} \cdot m\overarc{AC}$

4. Finding $m\overarc{AC}$:

Let $m\overarc{DE} = x$. From the problem, $m\overarc{AC} = 4x$ and $m\overarc{AD} = 120^\circ$. Additionally: $m\overarc{AC} + m\overarc{DE} + m\overarc{AE} = 360^\circ$

Substituting: $4x + x + 120^\circ = 360^\circ$ $5x = 240^\circ \quad \implies \quad x = 48^\circ$

Thus: $m\overarc{AC} = 4x = 192^\circ$

5. Calculate $\angle ABC$:

Using the inscribed angle theorem: $\angle ABC = \frac{1}{2} \cdot m\overarc{AC} = \frac{1}{2} \cdot 192^\circ = 96^\circ$

6. Verification:

Given the other conditions, $m\overarc{DE} = 48^\circ$ and $m\overarc{AD} = 120^\circ$, all values are consistent with the full circle.

The solution to the question asks for $\angle ABC$, which is 96°.

Would you like further clarification or step-by-step verification of any portion? Here are five related questions to expand on this:

1. How does the inscribed angle theorem relate to arcs subtended by tangents?
2. What is the role of tangents in calculating circle properties?
3. How would changing $m\overarc{AD}$ affect $\angle ABC$?
4. What happens if $m\overarc{AC}$ was smaller or larger?
5. Can this problem be solved using alternate geometrical methods?

Tip:

When solving problems involving circles, focus on the relationships between central angles, inscribed angles, and the arcs they subtend.