1. **Problem statement:** We have a circle with a chord subtending two angles $a$ and $b$ at points on the circumference and an inscribed triangle. The center of the circle is point $O$. We are asked to find the measure of the angle $\theta$ at the circumference where the chord starts, given the angle $\frac{4\pi}{3}$ at the vertex opposite the chord. 2. **Recall properties:** In a circle, the angle subtended by a chord at the center ($O$) is twice the angle subtended at the circumference on the same side. Also, angles on the same chord subtended on the circumference add up to half the angle at the center. 3. **Given angle at vertex:** The given angle at the vertex opposite the chord inside the triangle is $\frac{4\pi}{3}$ radians. 4. **Relationship of angles:** Since $\frac{4\pi}{3}$ is an exterior angle to the triangle or related to the central angle, we use the inscribed angle theorem: $$\theta = \frac{1}{2} \times \text{angle at center}$$ Since the vertex angle opposite the chord is $\frac{4\pi}{3}$, this corresponds to the central angle subtended by the chord. 5. **Calculate $\theta$:** $$\theta = \frac{1}{2} \times \frac{4\pi}{3} = \frac{2\pi}{3}$$ 6. **Answer:** The angle $\theta$ at the point on the circumference where the chord starts is $$\boxed{\frac{2\pi}{3}}$$ radians.