1. **Problem Statement:** Given a circle with center $O$, diameter $AB$, and points $C$, $D$, $E$ on the circumference such that $BC = BE$ and $\angle ADC = 120^\circ$, find the measure of $\angle COE$. 2. **Key Concepts:** - $AB$ is the diameter, so $\angle ADB = 90^\circ$ (angle in a semicircle). - $BC = BE$ means triangle $BCE$ is isosceles with $BC = BE$. - $\angle ADC = 120^\circ$ is an inscribed angle subtending arc $AC$. - Central angle $\angle COE$ subtends arc $CE$. 3. **Step 1: Analyze $\angle ADC = 120^\circ$** - $\angle ADC$ is an inscribed angle subtending arc $AC$. - The measure of arc $AC$ is twice the inscribed angle: $$\text{arc } AC = 2 \times 120^\circ = 240^\circ.$$ 4. **Step 2: Find arc $ABC$** - Since $AB$ is diameter, arc $AB = 180^\circ$. - Arc $AC = 240^\circ$ includes arc $AB$ and arc $BC$. - So, arc $BC = 240^\circ - 180^\circ = 60^\circ$. 5. **Step 3: Use $BC = BE$** - Chords $BC$ and $BE$ are equal, so arcs $BC$ and $BE$ are equal. - Therefore, arc $BE = 60^\circ$. 6. **Step 4: Find arc $CE$** - The entire circle is $360^\circ$. - Arcs $BC$ and $BE$ together are $60^\circ + 60^\circ = 120^\circ$. - Arc $CE$ is the remaining arc between points $C$ and $E$ on the circle. - Since $B$, $C$, and $E$ lie on the circle, arc $CE = 360^\circ - 120^\circ = 240^\circ$. 7. **Step 5: Find central angle $\angle COE$** - Central angle $\angle COE$ subtends arc $CE$. - The measure of a central angle equals the measure of the arc it subtends. - Therefore, $$\angle COE = 240^\circ.$$ **Final answer:** $\boxed{240^\circ}$